Euclidis Megarensis Geometriae Libri Duo Ab A M Severino Boethio Translati

THE TWO BOOKS OF THE GEOMETRY OF EUCLID OF MEGARA, TRANSLATED BY AN. MANL. SEVERINUS BOETHIUS.

BOOK ONE.

Since, my most practiced Patricius among geometers, at your urging I have undertaken to set forth and expound with a clearer approach the obscure pronouncements of Euclid concerning the figures of the geometric art, I think it must first be defined what measure is.

On Measure.

Measure is whatever is bounded by weight, capacity, length, height, breadth, and mind. The beginning of measure is called a point. A point is that of which no part exists. A line is length without breadth, and the ends of a line are points.

On the Kinds of Lines.

A straight line is one which extends evenly in its points. A surface is that which is reckoned by length and breadth. The boundaries of a surface are lines.

The Straight Line.

A plane surface is said to be that which is contained evenly within its straight lines. A plane surface.

On the Kinds of Angles.

A plane angle is the containment, in a plane, of two lines that touch each other and do not lie in a straight line with respect to one another.

But when the lines that contain the angle are straight, then the angle is called rectilinear.

When a straight line standing upon a straight line makes on either side the angles equal to one another, each of the equal angles is a right angle. And the line standing upon the straight line is called perpendicular. An obtuse angle is greater than a right angle, and an acute angle is less than a right angle.

On the Kinds of Figures.

A figure is that which is contained by some one or more boundaries.

A boundary is that which is the end of each thing.

A circle is a certain plane figure, carried round and contained by one line, which is called the circumference, to which all the straight lines that fall upon it from a point placed within the figure are equal to one another; and this point is called the center of the circle.

The diameter of a circle is a certain straight line drawn through the center and terminated on both sides at the circumference of the circle, which divides the circle into two equal parts.

A semicircle, however, is a plane figure that is contained by the diameter and the circumference which the diameter encloses.

Rectilinear figures are those that are contained by straight lines.

A trilateral figure is indeed one that is contained by three straight lines.

A quadrilateral, however, is one that is contained by four.

A finite measurable line is one that is determined either by some one of the observations or by some one of the terms.

A multilateral figure, therefore, is one that is contained by more than four sides.

On Triangles.

An equilateral triangle, then, is one that is contained by three equal sides.

An isosceles triangle, however, is one that has only two equal sides.

A scalene triangle, indeed, is one that has three unequal sides.

Further, of three-sided figures: a right-angled triangle is one that has one right angle.

An amblygonium, however, which in Latin is called obtusiangulum, is one that has an obtuse angle.

An oxygonium, that is, acutiangulum, is one in which the three angles are acute.

Concerning quadrilaterals.

But of four-sided figures, a square is called that which is equilateral and right-angled.

An oblong is that which is right-angled indeed, but not equilateral.

A rhombus is that which is equilateral indeed, but not right-angled.

A rhomboid, however, is that which has opposite lines and angles equal, yet is contained neither by right angles nor by equal sides.

Beyond these, however, all four-sided figures are called trapezia, that is, mensulae.

Parallels, that is, alternate straight lines, are called those which, placed in the same plane surface and produced in both directions, do not meet on either side.

Parallels.

On the petitions that are in geometry.

The petitions, which are called postulates (as pleased the ancients), are five.

The first postulates that from every point to every point a straight line be drawn.

The second admonishes that a defined straight line be produced into a continuous straight line.

The third instructs to designate a circle with every center and every interval.

The fourth requires that all right angles be equal to one another.

The fifth, however, if a straight line falling upon two straight lines makes the two interior angles on the same side less than two right angles, orders that the straight lines, produced indefinitely, meet on those sides on which the two interior angles are less than two right angles.

On the common conceptions of the mind that are in geometry.

The common conceptions, then, are those which are called by the Greeks: When spaces and intervals are equal to the same thing, they are also equal to one another; and if equals be taken from equals, the things that remain are equal; and if equals be added to equals, the wholes too are equal; and those things that agree with themselves are equal.

Every rectangular parallelogram is said to be contained by the two straight lines that enclose a right angle.

But of every parallelogram area, each of the parallelograms about the same diameter, together with the two complements, is called a gnomon.

Circles are equal whose diameters are equal; those that are not so related are unequal.

A straight line is said to touch a circle when, though it touches the circle, it does not cut the circle when produced in either direction.

Circles are said to touch one another which, touching each other, do not cut one another.

Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal to one another.

But that line is said to be farther from the center on which the longer perpendicular falls.

A segment of a circle is the figure contained by a straight line and the circumference of a circle.

An angle is said to be in a segment of a circle when some point is taken on the circumference of the segment, and from that same point two straight lines are drawn to the ends of the line.

The angle of a circle is said to be that which is contained by two lines drawn from the center. When the lines that are joined together enclose some small part of the circumference, the angle is said to consist in it.

A sector of a circle is the figure that is contained by two lines drawn from the center and by the circumference that is enclosed by those same lines.

Similar segments of circles are said to be those that receive equal angles, or in which the angles inscribed in them are equal to one another.

Therefore a figure is said to be inscribed within a figure when the figure that is inscribed touches the sides of the figure within which it is inscribed, each of its angles from the inner side.

But a figure is said to be circumscribed about a figure whenever the figure that is circumscribed touches, with all its sides, all the angles of that about which it is circumscribed.

HERE END THE PROLEGOMENA. HERE BEGIN THE DIAGRAMS.

On a given finite straight line to construct an equilateral triangle.

At a given point to place a straight line equal to a given straight line.

Given two unequal straight lines, from the greater to cut off a straight line equal to the lesser.

If two triangles have two sides equal to two sides, each to each, and the angle contained by the equal straight lines equal to the angle, they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles, each to each, those which the equal sides subtend.

The angles at the base of isosceles triangles are equal to one another.

If two angles of a triangle are equal to one another, the sides which subtend the equal angles will also be equal to one another.

On the same straight line, with the same two straight lines, two other straight lines equal, each to each, cannot in any way be constructed, toward another and another point, on the same sides, possessing the same ends as the first straight lines.

Of all two triangles whose two sides of one are equal to two sides of the other, and the base of one is equal to the base of the other, it is necessary that the two angles contained by the equal sides be equal.

To divide a given finite straight line into two equal parts.

From a point on a given straight line, to raise a straight line at right angles.

If two triangles have two sides equal to two sides, the one to the other, and have the base equal to the base, they will also have the angle equal to the angle which is contained by the equal straight lines.

Above a given infinite straight line, from a given point which is not on it, one must draw a perpendicular straight line.

Whoever, standing upon a straight line, makes angles, will either make two right angles or render them equal to two right angles.

If at any straight line and at a point on it two straight lines not drawn to the same side make the angles around themselves equal to two right angles, it is necessary that those lines lie in a straight line with one another.

If two straight lines cut one another, they make the angles at the vertex equal to one another.

In every triangle, when one side is produced, the exterior angle is greater than either of the interior and opposite angles.

In every triangle, two angles, taken together in every way, are less than two right angles.

In every triangle, the greater side is subtended by the greater angle.

In every triangle, the greater angle is extended under the greater side.

In every triangle, two sides taken together in any direction are greater than the remaining one.

If on any one side of a triangle, from the ends of that side, straight lines are drawn within and make an angle, the lines so drawn will be less than the remaining two sides of the triangle, but will contain a greater angle.

On a given straight line, and at a given point on it, to construct a rectilinear angle equal to a given rectilinear angle.

If two triangles have two angles equal to two angles, one to the other, and one side equal to one side, either the side adjacent to the equal angles or the side subtended by one of the equal angles, then the remaining sides will be equal to the remaining sides, one to the other, and the remaining angle will be equal to the remaining angle.

If a straight line falling upon two straight lines makes the alternate angles equal, the straight lines must be alternate to one another.

If a straight line falling upon two straight lines makes the exterior angle equal to the interior and opposite angle, or makes the interior angles on the same side equal to two right angles, the straight lines will be alternate to one another.

Through a given point, to draw a straight line alternate to a given straight line.

In every triangle, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

Straight lines that join equal and alternate straight lines toward the same parts of a straight line are themselves also alternate and equal.

Of those spaces which are contained by the sides of another and are called parallelograms, and whose opposite sides and angles are established equal to one another, the diameter also divides each into two equal parts.

All parallelograms that are established on the same bases and between the same alternate lines are proved equal to one another.

All parallelograms established on equal bases and between the same alternate lines are necessarily equal.

All triangles that are established on equal bases and between the same alternate lines are equal to one another.

Equal triangles that are established on the same base and on the same part are to be declared to be between the same alternate lines also.

Equal triangles established on equal bases and placed in a straight line, and on the same parts, must also be between the same alternate lines.

If a parallelogram and a triangle are established on the same base and between the same parallel lines, the parallelogram will be double the triangle.

Along a given straight line, a parallelogram equal to a given triangle must be set out at a given rectilinear angle.

One must place a parallelogram equal to a given rectilinear figure at a given rectilinear angle.

Of every parallelogram's area, the supplements of those parallelograms which are about the same diameter must necessarily be equal to one another.

A square must be described on a given finite straight line.

In those triangles in which one angle is right, which we call right-angled, the square that is described on the side subtending the right angle is equal to those squares that are written on the sides containing the right angle.

If the square described on one side of a triangle is equal to those squares which are described on the other two sides, the angle contained by the other two sides is right.

From the Second Book of Euclid of Megara.

If there are two straight lines, one of which is undivided, while the other is cut by any number of divisions, the rectangle contained by the two straight lines will be equal to the rectangles contained by the undivided line and each one of the divisions.

If a straight line is cut, the rectangle contained by the whole and one segment is equal to the rectangle enclosed by both segments and to the square described on the aforesaid segment.

If a straight line is cut at random, the square written from the whole is equal to the squares described from each segment, and corresponds to twice the rectangle contained by the same segments.

If a straight line is cut both into equal and into unequal parts, the rectangle contained by the unequal sections of the whole, together with the square described from the line between the two sections, is equal to the square described from the half.

If a straight line is cut both into equal and into unequal parts, the squares described on the unequal portions of the whole are double the squares that are made from the half and from the line between the two sections.

If a straight line is divided into equal parts, and another straight line is joined to it in a straight line, the rectangle contained by the whole with the added line and by the added line is equal to the square described from the half together with the square described from the line consisting of the added line and the half.

If a straight line is cut into equal parts, and a certain straight line is joined to it in a straight line, the square that is described from the whole with the added line.

Both squares taken together are necessarily double the square described from the half and the square described from that which consists of the half and the added line, both squares taken together.

It is required to cut a given straight line in such a way that the rectangle contained by the whole and one portion is equal to the square made from the remaining section.

In this triangular figure which has an obtuse angle, the sides subtending the obtuse angles are as much greater in square than the side containing the obtuse angle as is twice the rectangle contained by one of the sides about the obtuse angle, on which the perpendicular when produced falls, and by the segment that is found outside from the perpendicular to the obtuse angle.

It is necessary to place a square equal to a given rectilinear figure.

FROM THE THIRD BOOK OF EUCLID OF MEGARA.

If in a circle a certain line is drawn through the center, and it cuts a certain straight line not placed at the center into two equal parts, it cuts it at right angles. And if it cuts it at right angles, it divides it into two equal parts.

In equal circles, the angles that stand on equal circumferences are equal to one another, whether they are constituted at the centers or at the circumferences.

It is possible to divide a given circumference into two equal parts.

In a circle, indeed, the angle that is in a semicircle is a right angle; but the angle that is in the greater segment is less than a right angle. The angle that is in the smaller segment is greater than a right angle, and the angle of the greater segment is greater than a right angle, while the angle of the smaller segment is less than a right angle.

If a straight line touches a circle, and from the point of contact a straight line is drawn through the circumference cutting the circle, whatever angles it makes, two angles in alternate segments of the circle are equal.

From this, therefore, it is clear that if from a point on a circle two straight lines touch each other and are equal to each other, it is proper to describe parts of a circle upon the given straight lines.

FROM THE FOURTH BOOK OF EUCLID OF MEGARA.

Within a given circle, it is required to fit a given straight line, which is not greater than the diameter, as an equal straight line.

It is fitting to place, within a given circle, a triangle of equal angles to a given triangle.

Around a given circle, a triangle of equal angles to a given triangle is to be marked out.

It is necessary to design a circle within a given triangle.

It is useful to describe some square within a given circle.

To design a circle within a proposed square.

Geometers prescribe to design around a given circle an equilateral and equiangular pentagon.

It is not unfitting to design within a given circle a pentagon that is equilateral and equiangular.

For all things whatsoever that exist by reason of numbers stand together by their own nature. And proportionally, some are constituted from others by equality of circumference, exceeding by their own multiplications, and alternately making their limit by their proportions.

On geometrical figures.

Therefore, since the speculations of the figures set out above were published succinctly and obscurely by Euclid, and translated by us briefly, expressing word for word as it were, certain things seem to need to be repeated and gone over again, so that the mind of the reader may not be deterred by obscurity, but rather delighted by us with the light of some example poured in. For certain things must be inserted by us into this work, very necessary for this art, both corresponding to the things said above and suited to those that follow. And whoever approaches the understanding of these things instructed in the theorems of our Arithmetics is led by a readier understanding.

It has therefore been stated above that it is necessary to construct an equilateral triangle upon a given finite straight line, but too obscurely; wherefore we have appended a note of this example. Let the given finite straight line be A B; it is necessary, therefore, to construct upon A B an equilateral triangle. With center A and radius A B, let the circle B C E D be drawn. And again, with center B and radius A B, let the circle A C F D be drawn, and from the point G, where the circles divide each other, let lines be joined to the points A and B, namely the straight lines C A and C B. Since, therefore, the point A is the center of the circle B C E D, A B is equal to A C. Again, since the point B is the center of the circle A C F D, B A is equal to B C. But A B has also been shown to be equal to C A, and therefore it will be equal to B C. Therefore the three lines C A, A B, B C are equal to one another; therefore the triangle C A B is equilateral. And so upon the given finite straight line A B the triangle has been constructed, which was what was required to be done.

In the preceding passages it was indeed stated that it is necessary to place a straight line equal to a given straight line at a given point. But this is obscure and difficult for those unskilled in this art. Therefore, delighting the reader's mind as if by way of introduction, we make clear the explanation of the following figure by setting out the marks of the letters and lines. Let the given point be A, and the given straight line B C; it is required, then, to place at point A a straight line equal to the straight line B C. For let the straight line A B be joined from point A to point B, and upon the straight line A B let the equilateral triangle D A B be constructed, and let the straight lines D A and D B be extended in a straight line to A G and B M, and with center B, at the distance B C, let the circle C F E be described, and again with center D, at the distance D E, let the circle F K L be described. Since, therefore, point B is the center of the circle C F E, C B is equal to B F. Again, since point D is the center of the circle F L K, D L is equal to D F. Of these, D A is equal to D B, for the triangle D A B is equilateral. The remainder A L therefore is equal to the remainder B F. But B F has also been shown to be equal to B C. And B C will therefore be equal to A L. At the given point A, therefore, a line equal to the given straight line B C has been placed, namely the line A L, which was required to be done, as the figure set out below shows.

In the third place, therefore, Euclid set forth above that it is fitting, when two unequal straight lines are given, to cut off from the greater a straight line equal to the lesser, but too briefly, and for that reason obscurely and confusedly. We, however, so that the mind of the reader may be led, as it were by certain steps, to the approach of clearer understanding, have appended the description of this proposition in a formula. Let two straight lines be given.

Unequal, A B and C D, and let A B be the greater. It is required, therefore, to cut off from the greater A B a line equal to the lesser C D; for let a line A E equal to C D be placed at the point A. And with center A, and with the space A E, let a circle E G F be described; since, therefore, the point A is the center of the circle E G F, A E is equal to A G. But C D was also equal to A E, and C D will be equal to A G. For two unequal straight lines having been given, namely A B and C D, from the greater, which is A B, there has been cut off A G, equal to the lesser, which is C D, which it was required to do.

Once the reader has been initiated by these brief introductions, which are yet altogether indispensable to beginners in this art, if he wavers and shrinks back from some of the things set forth above, he can by himself discover and compose similar descriptions of figures without any objection or hindrance.

But now it is time to come to the tradition of the geometrical table, adapted into Latin by Archytas, no mean author of this discipline, if I first set forth what the kinds of angles and lines are, and say a few things about summits and extremities.

The kinds of rational angles, then, are three: right, obtuse, and acute, and these have nine species: three of straight lines, three of straight lines and circumferences, and three of the obtuse and circumferences.

A right angle is the orthygrammos, that is, one comprehended by straight lines, called in Latin normalis. But whenever a straight line standing upon a straight line makes equal angles, and a perpendicular line is joined, it will produce a right-angled triangle.

An obtuse angle is greater than a right angle, that is, it exceeds the position of a right angle, because even if a triangle is constituted according to this position, the perpendicular will be had outside the bounding lines.

An acute angle, however, is more compressed than a right angle; if from a straight line, as if from a base, it sends forth a straight line according to its own inclination, and by a like compression receives a straight line in meeting, it will make a triangle which has the perpendicular within the three lines.

The kinds of lines are three: straight, circular, and curved.

A straight line, then, is one which is placed evenly in its points, which, placed evenly in a plane, does not meet. A circular line, however, is one whose points, curved on both sides and distant from one another, do not meet; which points, if they come together, the circle should not be called a circular line. A curved line, however, is one that is multiform, like trees or rivers, and the rest of the points, in whose likeness the boundary of the most narrow fields is also bounded, and of many things which are similarly formed in an even line by nature.

Therefore there are two kinds of boundaries: a boundary and a plane boundary. A boundary, according to the geometric designation, is that which extends in length and breadth.

The limits of a boundary are lines.

A plane boundary, however, is that which is bounded on every side equally by straight lines.

Of all boundaries, moreover, there are two observations in enclosing: the rigid and the flexible.

The rigid is that which is contained on every side by straight lines.

The flexible, however, is that which, for the sake of lessening labor and while preserving the principle of right angles, is stretched according to its own extremities.

For there are two kinds of extremities: one which is kept for rigidity, and the other for flexibility. Rigidity is whatever is seen directly between two points, as it were in the manner of a line.

The flexible, however, is whatever is curved according to the nature of places. For what in a field has been directed to an end by a reaper for the sake of the work is called rigidity; whatever is written in the form of an I in imitation of these things is called a line.

There are two kinds of rigor, when they extend through the individual intervening spaces, as roads generally run.

But one who would understand this art must know what digits are, what articles are, what composite numbers are, what incomposite numbers are, what multipliers are, and what divisors are, for the speculation of this form which we are about to hand down. The ancients were accustomed to call digits all those below the first limit, that is, all that we count from unity up to the sum of ten: 2, 3, 4, 5, 6, 7, 8, 9.

But all the articles placed in order from ten onward and progressing to infinity are called composite, since they are all numbers from the first limit, that is, from ten up to the second limit, that is, the rest following in order, with the limits excepted. Incomposite, however, are all the digits together with the enumerated numbers and all the limits.

Multipliers, then, are numbers that are turned about by mutual replication upon themselves, that is, sometimes the greater is the multiplier of the lesser, sometimes the lesser of the greater. Sometimes, however, a number growing into itself receives the increase of multiplication. Divisors, moreover, are always established as lesser numbers of greater ones.

On the Method of the Abacus

Therefore the men of ancient prudence, following the Pythagorean doctrine, and the curious investigators and observers of Platonic authority, established the whole summit of philosophy by the power of numbers. For who, by judging, could thoroughly know the melodies of musical harmonies without numbers? Who, ignorant of the nature of numbers, could grasp the starry bodies of the firmament itself, compacted of stars, and gather the risings and settings of the signs?

But concerning arithmetic and geometry, what need is there to speak, since if the force of numbers perishes, it does not even appear in naming? Because enough has been said about this in arithmetic and in music, let us return to what must be said.

But the Pythagoreans, lest they ever be deceived in multiplications and divisions and in calculations by foot, as they were most ingenious and subtle in all things, described for themselves a certain formula which, in honor of their teacher, they called the table of Pythagoras, because what they had depicted they had learned from their master showing it beforehand; it was called by later writers the Abacus, so that what they had conceived in their lofty mind they might be able to pour into the knowledge of all, better if they showed it as if by seeing; and they formed it with the rather wondrous description given below.

But they used the formula of the arrangement set out above in this way. For they had variously shaped tops or characters. Some had written down for themselves notations of tops of this kind, so that this little mark corresponds to one: 1; that one to two: 2; the third to three: 3; the fourth to four: 4; these to five: 5; that one to six: 6; the seventh would correspond to seven: 7; these to eight: 8; those to nine: 9. But some, in the depiction of this form, adopted them like the letters of the alphabet, in such a way that the first letter was for one, the second for two, the third for three, and the rest in natural order obtained marks and inscriptions of number only. For they were accustomed to scatter these tops, thus variously like dust, in multiplying and in dividing, so that if they placed the characters already mentioned under one, joining them in the order of natural number, nothing other than digits would arise. But placing the first number, that is two—for one is not a number, as is said in arithmetic, but the fount and origin of numbers—as 10 written, and three as 30, and four as 40, and the rest following in order, they established that they assign their own according to their denominations. But under the line marked with the number one hundred, placing the same tops, they decreed that two be 200, three 300, four 400, and the rest correspond to fixed denominations.

But in the following lines of the little pages, doing the same, they are obscured by no cloud of error.

It must be known, however, and examined with careful scrutiny, in multiplying and dividing, to which little page the digits and to which the articles are to be joined. For a single-digit multiplier [places] ten digits in the tens, and articles in the hundreds. The same single-digit multiplier [places] a hundred digits in the hundreds, and articles in the thousands. And the multiplier of a thousand [places] digits in the thousands, and articles in the ten-thousands; and the multiplier of a hundred-thousand will have digits in the hundred-thousands, but articles in the ten-thousands. The ten, however, as multiplier of itself, [places] digits in the page inscribed, and articles in the thousands; and the multiplier of a hundred [places] digits in the thousands, and articles in the tens; and the multiplier of a thousand [places] digits in the tens, and articles in the hundreds; and the multiplier of a hundred-thousand will have thousands. The hundred, indeed, equally as multiplier of itself, [places] digits in the tens and articles in the hundreds and thousands. Multiplying digits in the hundreds and articles in the ten-hundreds; and the hundred-thousand multiplying, digits in the ten-thousands and articles in the hundred-thousands; and the ten-thousand multiplying, digits in the thousands and articles in the ten-thousands, they will set beneath. The thousand likewise multiplying itself, digits in the tens and articles in the hundreds.

And the hundred-thousand multiplier is recognized to have digits in the hundred-thousands and articles in the thousands, and by making a ten-thousand grow, digits in the ten-thousands and articles in the hundred-thousands. But the ten-thousand multiplier, multiplying a hundred-thousand, is found to have digits in the hundred-thousands and articles in the ten-thousands likewise, and adding itself to itself, digits in the hundred-thousands and articles in the thousands. But the hundred-thousand, multiplying itself, likewise sets down digits in the ten-thousands and articles in the hundred-thousands.

Concerning divisions, a heading.

Divisions, then, however many, the mind of the reader having now been introduced, it is able to discern well; for we are about to speak briefly of these and only on the surface, and if any obscure points intervene, we leave them to the diligent exercise of readers to investigate. If a ten, or a hundred, or further numbers are proposed to be divided by themselves, the lesser must be subtracted from the greater until they divide; but a single divisor of ten and a hundred, or of a thousand, or of further numbers, or a ten as divisor of those following it, ought to divide them by the difference taken. But a ten compounded with a unit is to be divided by seconds or thirds. And thereafter, according to the denomination of the parts, a ten, whether simple or compound, is to be divided.

But one who is a diligent investigator will recognize that the hundred-thousand, or higher numbers composed by tens, can be divided if he takes the difference and divides by the first articles, or by the seconds set in place, and by dividing the increased by the underlying. But the hundred composed with single M's is known to divide the hundred or the thousand in this way. Therefore, having taken one of the things to be divided, what remains must be made equal to the divisor, and what is left over must be set aside and reserved. But the single, or as others wish, the unit, must be multiplied by the equating of the greater, and indeed the difference of the perfect must be placed under the digits, but the imperfect of the articular must be placed before, and the whole must be joined to the previously set aside. And this difference, even if perhaps some one has been shut off, has signified what is left from the things to be divided. Having tasted these things in a brief introduction, if any things are said obscurely, lest they be passed over to tedium, we commit them to the exercise of the diligent reader, making the end of this book, and as if turning ourselves to the further things of those that follow.

BOOK TWO.

In the treatise of the preceding volume, all the theorems of the geometric art have been stated, albeit succinctly. But this book, by unraveling it, will unhesitatingly settle the knowledge of measurements, which is the subject of inquiry, and the scrupulous subtleties of all measurement-related questions. For the ancient land-surveyors were accustomed to make every square measure of a field longer or wider by half. And what was longer in width they wished to call a bench, and what was longer in length the same, as the formula set out below teaches.

On the heading of measures.

The ancient and most careful examiners of measurement therefore established twelve kinds of measures, with which, when they wished, they would measure out the areas of shapes and fields. Their names are these: the mile, the stadium, the actus, the ten-foot rod, which is also the perch, the pace, the step, the cubit, the foot, the half-foot, the palm, the inch, the digit. Now it is ordained that the mile have a stretch of a thousand feet. The stadium, moreover, is established to have [a fixed number of] feet. The actus is divided three ways: into the smallest, into the square, into the doubled. The smallest actus extends only four feet in width and [a fixed number of] feet in length. But the square actus is enclosed on every side by feet. The doubled actus, however, unfolds [a fixed number of] feet; the ten-foot rod collects ten feet; the pace [a fixed number]; the step and a half. The cubit is recognized to have one and a half feet. The foot has four palms, the half-foot two, and the palm is completed by the stretch of four digits.

But concerning the uncial and digital measure, shall we speak better when we have discussed uncials, both their marks and names, in what follows? And we shall speak more clearly when we have promised concerning the subtleties of the elements and the minutes; now it is fitting for us to return to the narration of the following treatise, if we have first demonstrated what the foot is when extended, what when contracted, and what when squared. For the foot is called extended when only the measure of a foot is recognized in length. But that foot is judged contracted in which both length and breadth are considered. And the foot is held to be squared when the consideration of three dimensions is assessed by inequality; but now it is time to approach what we have undertaken.

On measure and the three dimensions, a heading.

Although also in the beginning of the preceding book we designated what measure is, nevertheless it pleases us to satisfy the observers of this art in a special way, according to Julius

What is a measure, as Frontinus, the most prudent inspector of the geometric art, defines it.

A measure, indeed, is the finite length of several intervals equal among themselves; but the theoretical consideration of the measuring geometric art is gathered from the consideration of three dimensions, that is, of length, width, and thickness. And so that it may be resolved more clearly, it is distinguished as straight, plane, and solid. A straight [measure] is that which is reckoned by measuring length alone, as the lines of a portico, the distances of miles, the widths of rivers, and very many other things stretched out in a direct long extension, as the diagram of the line depicted below indicates.

Straight.

A plane is that which the Greeks call epipedon, but we call it measured in feet, because it is considered through length and width, as the flatness of fields, and the areas of buildings without roof-work and ceilings and paneling and the like, as the formula below teaches.

A solid is also what the Greeks call stereon, but we call it square feet, because it is proved to have both length and breadth and thickness: as of buildings, pillars, and pyramids, and also of heaps of stones, and many other things, as the formulas set out below note.

Solid.

On the reckoning of measurements by feet.

But now it is time, as we promised, to touch upon the knowledge of problems concerning measurements by feet by narrating it, and to speak at once of the investigation of the reckoning of feet. Concerning triangles, indeed, which, just as the ternary naturally precedes the quaternary, are for that reason to be placed before tetragons and pentagons and the rest, I judge that it must be said first.

On triangles.

Now there are six principal kinds of triangles: isopleuron, isosceles, scalene, right-angled, obtuse-angled, acute-angled; the forms and measurements by feet of all of these we shall explain in what follows.

On the isopleuron.

The isopleuron triangle, therefore, which was called the equilateral triangle almost at the beginning of the preceding book, is proved to have equal sides. Let there be set out, then, an isopleuron having thirty feet on each of its sides; its embadum, that is, its area, must be investigated in this way. For the sum of one side multiplied by itself completes a number; if five hundred and one are subtracted from these, 399 are left; this triangle of the isopleuron kind contains an embadum of that many feet. For its perpendicular is established to extend to [a certain number of] feet.

If they have grown by being multiplied by half of one side, that is by 15, they fill the area; or if a third part of one side, by three and ten, it will be increased; but if they multiply the sum of one side by the same three, they will return ninety, which, joined to the above 300, will make 390, that is the area of the aforementioned triangle. Let there be, moreover, the depiction of the aforesaid matters made below.

But so that the reader be not hindered in investigations of this kind by any cloud of error and ignorance, we shall therefore append an example of the manifestation of the same equilateral triangle, that is of a solid with equal sides. Come now, let it be equilateral, whose individual sides comprise feet. If one of these, increased by itself, has grown, a sum will arise. And if you have added to it the number of one side, 8 and 2 will be born; by the half written above of these you will mark out the area of the aforementioned equilateral triangle, as the formula of the description set below teaches.

But of this triangle so often mentioned, so that anyone who inquires may be able to investigate the measure of each side and state it, we shall give a most clear demonstration of the method. Let it therefore be proposed, if they determine the area to be extended in feet, that it be known how many square feet each side comprises. Let the above-written area therefore be multiplied by eight, and it rises to a number; if one be added to this, they are made; if I take the side of this sum, it will be fifty-seven. If a unit be subtracted from this, they are left. If I investigate the half of these, they will be. Thus therefore each side of the equilateral triangle is extended in feet.

But the isosceles triangle, which was determined by Euclid, most skilled in geometry, having two sides equal, is established second in order among triangles. Let its two sides be extended by unequal numbers, namely 25, and let the base be known to have a space of fourteen feet. It remains therefore that we inquire how many feet the height of the area comprises. For if the half of the base, that is 7, be multiplied by itself, they will be born. But the measure of one side, if you multiply by itself, that is 23, you will return, from which if you set aside, they are left. If you take the side of these, they will be, and by so many feet the height of this triangle is established to be extended. But how many feet the area has, it must be done thus that it be found. The half of the base must again be taken, that is; which if you multiply by the height, that is by 24, you will make 168, of so many feet is the area of the above-mentioned triangle.

Basis 14

On the heading of the scalene triangle.

The scalene triangle, then, is defined by Euclid as having three unequal sides. But we shall give clear examples of its numbers, and let a scalene triangle be proposed, which the Latins call a wedge, whose shorter side's slope comprises feet, and let its base be noted to have linear feet. But how many feet the perpendicular and the area of this triangle extend to, it remains to be sought. Let the sum of the shorter side therefore be multiplied by itself, and they make 220. Likewise, if the base is multiplied by itself, they grow, and when these are combined into one, they are born. This, therefore, if by moving it aside it is fitting that the sum of the longer side be multiplied by itself, which multiplication brings forth a number. And indeed, if you subtract this number from the sum previously set aside, namely from 850, they remain. If you take the half of these, you will unfold 225; from which, if the sum of the base or the feet is taken away, it will be nine — by so many feet the cutting-off, or the lesser projection, of this triangle is contained. It remains that it be asked how many feet the perpendicular has. Let the lesser side therefore be multiplied by itself as above, and 25 comes forth. Again, also, the increased sum of the lesser cutting-off is multiplied by itself, and it produces; if you take these away from the side multiplied by itself, 144 remain. Of these the side is recognized to be twelve; the perpendicular of this triangle is said to comprise so many feet. But the measure of the area is found in this way. Let the perpendicular therefore measure the base, or 12, 20; they will rise.

By half of these the area of the often-mentioned scalene triangle is measured in feet, as is noted in the figure below.

On the heading of the right-angled triangle.

In the fourth place, namely, the right-angled triangle is inserted by Euclid, and it is defined as having one right angle and containing unequal sides; we judge it to be more difficult and more obscure than the others, even for those who hear it. And for this reason we shall make a longer delay in its explanation. Let there be, then, a right-angled triangle, whose cathetus is extended marked by an equal number, or measured in species. If its sides are unknown, they are ordered to be investigated in this way by Archytas. Let the half, then, of the aforementioned cathetus be taken, that is 4, and let it be multiplied by itself, and 16 will arise. If unity is subtracted from these, 15 will appear. The base of this triangle is known to be of so many feet. But to the sum of the aforesaid increased by half of the cathetus, if one is added, there will be 17 feet of the hypotenuse. By the same sum, that is by 16, the area is to be found; therefore let the middle of this sum be multiplied by the cathetus, and they will arise, which fill up the reckoning of the areas, which is clearly declared in the figure below.

On the same heading.

Let us therefore attempt to set out a clear and certain description of this right-angled triangle by the quantity of even and odd number. Let an even number, then, be assigned first to the cathetus, that is 6; by half of it multiplied by itself, they are produced. If from this, according to the rule of our precept designated above, one be taken away, it will be the eight, the base of this triangle; half of this, namely the four, multiplied by the cathetus, completes the area, according to what was said above; the hypotenuse is sought through the cathetus and the base in feet without any objection. We shall give an example of our authority, very easy and clear to state; for let the half of the cathetus of this triangle be multiplied by its own quantity, and to the sum which has come from this multiplication let unity be added, and it will be the measure in feet of the hypotenuse; but if one be taken away from the same, it will be the base. And let this be the description made of this matter.

Let us therefore establish a probable explanation of this right-angled triangle by means of odd number. Let the cathetus be marked by an odd number, that is, one which, if you multiply it by itself, you will resolve, from which, when unity has been subtracted, eight remain; if the half of these is taken, it is proved that it will be the measure in feet of the base of this right triangle.

But to this base, or to its half, or to 4, if you add one, you will prove it to be the hypotenuse of the triangle; let the area be found, as was said above, that is, let the cathetus be increased by the half of this base, as is seen in the figure below.

But lest any obscurity deceive the curious investigator of this discipline, let it not displease us to discuss this same right triangle again. For there is another method of finding the cathetus and the base and the hypotenuse. Let the cathetus therefore be set out extended in feet; if you multiply it by its own quantity, you will note it; but let the base having 12 feet be inscribed, which, if it has grown together by itself like the cathetus, will be produced.

Those sums, that is, also combined, will restore 169. Their side is manifest, that is, the hypotenuse of the aforementioned triangle. Finally, if you multiply the hypotenuse by itself, you will produce an amount equal to the combined quantity, that is, 169. If from these you subtract the cathetus multiplied by itself, the remainders are those whose side, that is, 12, restores the base. From the hypotenuse multiplied by itself, if one subtracts the base multiplied by itself, that is, from 169, [the remainder is] no more than what remains. The side of these, that is, 5, constitutes the cathetus; and the area is measured by the half of the base and the cathetus multiplied together.

Likewise, it is fitting in this triangle to declare the measure of the base by means of the cathetus. Let the cathetus above be 5, and this multiplied by itself constitutes [25]. If I take away one from this, there will proceed [24], the half of which makes the base. Again, if I add the same unity to the quantity of the base, I will unfold the hypotenuse; but if the base is multiplied by the cathetus, 60 proceeds as the sum. The half of these completes the area.

Likewise, a heading concerning the same matter.

Furthermore, for one who is advancing step by step from this point, we wish to set before him another rule of this triangle by way of an example, so that those who investigate it cautiously may most cautiously be led by the line of the method without doubt to that which they desire. Let there be set forth, then, a description of the same right triangle with the same quantities by which it was defined above, namely the perpendicular 8, the hypotenuse 17, and the base defined in feet. But now let us endeavor to demonstrate by what reasoning the sum in feet of the perpendicular and the base can be found through the measure of the hypotenuse. Let us therefore multiply the sum of the hypotenuse by itself, and the number 289 results. If from this four times the quantity of the side is subtracted, there remain [blank]. If you seek the square root of these, you will find it to be [blank]; namely, if you add this 7 to the perpendicular and base combined, you will make [blank], half of which constitutes the measure of the base; but if you subtract fifteen from the combined total, that is, from 50, you will prove without doubt that the perpendicular remains.

Likewise, a heading concerning the same matter.

Let us again designate the form of the aforementioned right triangle from other quantities of numbers, so that when someone attempts to hand down the open teaching of this triangle either through a larger or through a smaller number, he may slip into no error. Come now, let there be a right triangle which a smaller and two odd numbers surround: let an even number be assigned to the base, or one odd number to the perpendicular, that is 15, and the other to the hypotenuse, that is 25. But the conclusion of the area is to be sought according to the rule of our aforementioned precept, that is, by the multiplication of half the base and the whole sum of the perpendicular; for it contains seven hundred fifty contracted feet of area. But the perpendicular and the base are to be investigated by such a method: let the sum of the hypotenuse be multiplied by itself, and it comes to 625; to which if four times the area be added, 1225 will arise; whose square side, that is 35, if you take out, you will confirm both sums, of the base and of the perpendicular.

But one must know and investigate by what number the perpendicular and the base differ from one another. Let us make clear what this is. If, then, I multiply the hypotenuse by itself four times, and subtract the embada which I added above, it returns to the sum; a fifth part of these holds the difference, that is, 5. If again I join this to the two combined sums, namely 20 and 15, the middle of these completes the base.

But if I take away the same difference, that is, 5, from the base, I establish the perpendicular, as can be seen in the figure below.

A heading concerning the circle inscribed in a right triangle.

Now also, what was approved by the judgment of Archytas in this same right triangle, and was first reasonably discovered by the most diligent investigation of Euclid, we have deemed it worthwhile not to pass over. For it often happens that a disputant in geometry asks, if a circle is inscribed in this right triangle, how many feet the diameter comprises; and lest anyone, overcome by ignorance, say he refutes this, we briefly set out the matter as follows: let a circle, then, be inscribed in the right triangle, touching all its lines; this being done, the perpendicular and the base are gathered into one; from the combination of this sum, if you take away the quantity of the hypotenuse, you will make the diameter; for the joined 8, that is, the perpendicular and the base, yield it. From these, if I subtract the hypotenuse, that is, 15, I establish that I obtain the diameter 0, as the figure made below designates.

On the rubric for the obtuse-angled triangle.

Fifth in the order of triangles, the obtuse-angled triangle inserted by Euclid is said to have an obtuse angle, which we undertake to explain briefly and clearly. For if the diligent reader comes instructed by the precepts and formulas of our preceding demonstration, he will not stumble at all in this. Let there be set out an obtuse-angled triangle whose base is marked by the number 18, the greater hypotenuse by 10, and the lesser by 9; the perpendicular is marked by the sum 1.

Let the base, then, be multiplied by half the perpendicular, that is, 18 by two, and they come out; by this sum the plane extent of the areal space is filled. But Archytas, using reason in all things, established that the area of this obtuse-angled triangle is found in another way; we judge that this sum is not taken in this areal extent, but that a lesser can be contained. For he asserted that the perpendicular must be multiplied by itself and by two, or by itself and by eight, exceeding itself by double, and that the quantity which came from this multiplication constitutes the area—not as 36, but that areal contemplation gathers it into itself. But whoever wishes to designate the shapes of the aforesaid triangle on a plane, let him begin the matter from the quantity of the base in this manner, so that the term of the lesser and the greater hypotenuse joined together exceeds the term of the base by a little; that is, if the base is measured by feet, the greater hypotenuse is marked 11, and the lesser likewise. But we shall show this better by what we have set out in numbers, if we subject the form of some example.

On the rubric for the acute-angled triangle.

It remains for us to speak of the acute-angled triangle, which Euclid, no insignificant geometer, places sixth in the description of triangles, defined by its acute angle. Let there be, then, an acute-angled triangle, whose shorter side's limit, that is the lesser hypotenuse, is bounded at 13 feet, while the greater side and the base is measured at 1; whose sum of cathetus and embadon, if unknown, shall be gathered by the following reasoning.

Let the quantity of the shorter side be squared, and 169 results; likewise if the limit of the base is multiplied by itself, they are produced; if you join these into a sum, you will effect it. This done, let the term of the hypotenuse also be multiplied by itself, and a number will arise; if I separate this from the sum combined above, 140 remain; let the half of these be noted, which, divided by the base, contains it five times in itself. But the lesser segment obtains the sum of this dominion, which, multiplied by itself, establishes. If you subtract these from the sum of the shorter side multiplied by itself, there remain, whose tetragonal side, which is, will fill out the sum of the cathetus. Take care to investigate the conclusion of the area in this way: draw the middle of the base through the cathetus, that is through 12, and 85 will result. Do not be ignorant that the area's pavement of this triangle completes this sum; let therefore a figure of this kind be described.

But since in the preceding pages we have explained the consideration of the foot-measure of triangles for the investigation of diligent readers, it remains for us to make a concise transition to the contemplation of quadrilaterals, being about to produce a treatise on these matters.

For the collection of squares is easier than that of the others, and first indeed let us begin concerning the normal quadrilateral in this manner. Every quadrilateral, therefore, regularly constituted, by multiplying breadth by length, establishes an areal plane and undoubtedly completes its foot-measure. Let a quadrilateral now be set down, marked by an even number, that is 8, which, multiplying breadth by length through itself, I shall make—namely the area of the quadrilateral described below.

But if you do the same by an odd number, the same reasoning will stand firm, carefully against the obstacle; this normal quadrilateral is called by Euclid equilateral and right-angled, and by Nicomachus likewise in his Arithmetics it is similarly named.

On the longer side, a heading.

But the rectangle with one side longer is defined by Euclid as right-angled but not equilateral; by Nicomachus, however, it is called heteromfres. Its length multiplied by its width demonstrates the area's total measure in feet, whether the terms be even or odd, provided only that it be a rectangle with one side longer, whose length is 8 feet and width 4, or length 9 and width 6, or 5, or 4, it collects.

Let the width therefore multiply the length, that is 8; they will be born, that is, the area of the rectangle with one side longer will come forth, which these figures of numbers designate by even and odd number alike.

On the rhombus, a heading.

To these already-mentioned parallelograms we think rhombuses and rhomboids should be added; for although they differ from the above-mentioned parallelograms either in angle or in side, they are nonetheless to be counted among them. Come now, let there be a four-sided rhombus, written down with each of its sides measuring ten feet, but let its diagonal, that is the direction of the angular line, be reckoned at twice six in quantity; its 6, if multiplied by itself, will arise; which if you subtract from the square of the base multiplied by itself, there remains. The side of these, that is 8, constitutes the perpendicular of this rhombus. But the diagonal drawn through the perpendicular shows the space of the area's total. This is defined by Euclid as having equal sides, but not equal nor right angles.

Let this, then, be the development of this form.

On the rhomboid, a heading.

Euclid, indeed, defining the rhomboid, set it forth as having neither equal angles nor equal sides; we too lay it open by forming it with a more accessible approach and by ascribing numbers. Come now, let there be a rhomboid whose one side is 4 feet, the second 4, the third 6, and the fourth 2; but if you join the largest terms of these sums, which obtain the length, you will make 14, whose half the number seven constitutes. But the smaller sums, reduced into one, complete the quantity six, whose half the number three fills out; and these, that is the halves, if multiplied by themselves, will rise to 21, that is the areal feet of this tetragon, as appears below.

For to these things we think it not unfitting to add the right-angled trapezium, marked out on its sides by the double and sesquialter proportion of numbers. Let the sum fifteen be ascribed to the top, but thirty to the perpendicular, exceeding also by double, while to the base let a term be assigned preserving toward this the sesquialter relation; by these sums, therefore, the area of this trapezium is to be established in such a manner. Let the top be joined to the base, that is 10 and 60; the term abounds. If its half is multiplied by the perpendicular, it discloses the extent of the area, as is plain in the figure written in the middle.

On finding the diagonal, a heading.

It often happens in the contemplation of this art that the extension of a diagonal line is required, namely one that obtains the feet of these quadrilaterals. So that this not be unknown, we shall give a very easy and most clear example of this reasoning. Let there be placed a parallelogram of 60, right-angled, in length 80, and having in height 60 feet; the length multiplied by itself six times unfolds, while the other multiplied by itself three times produces, namely six times 400, and three times 600, brought together into one sum, restore IV; if I take the square side of these, I will mark it out, that is, the diagonal of this right-angled parallelogram, as can be seen in the figure written within.

On multi-angled figures, a heading.

But since we have spoken sufficiently and briefly about the ratios of quadrilaterals, it remains that we discuss pentagons and hexagons and the rest. Every pentagon, then, having equal sides, with the sum of one side increased in itself and taken three times, and again the same subtracted, and half of this sum taken, discloses the surface of its areal space. Let there be in this way a pentagon having each of its sides six feet; if I multiply these by themselves, I will restore; I will note these taken three times into a number; if I subtract from it the sum of one side, that is, six, I will complete; if I take half of these, I will fill the area of the pentagon described below.

On the hexagon, a heading.

The hexagon is next to be discussed in order; let a hexagon be described with its sides marked, namely the one that I make by multiplying eight by itself; this sum, that is, taken four times, is left over; to these, that is, let the quantity of one side, namely twice taken, be added, and 212 appears. If you take the middle of these, you will unfold the area of this hexagon.

On the heptagon, a heading.

After these things, that we may set forth the reckoning of the heptagon that follows, it is fitting that this heptagon, namely seven, be inserted here in the third place, just as it appears naturally placed third in order among the odd numbers; for the heptagon is circumscribed by the quantity six, and if you multiply the sum of one side by itself, you will note it, namely that quantity, that is, 36, taken five times, contributes to what is present. If from these you subtract the quantity of six taken three times, 162 is left; the half of these, taken as 81, brings it about that you have the area of this heptagon.

On the octagon, a heading.

The octagon, indeed, being established fourth in the natural order of even numbers, is to be taken up naturally fourth in this place of discussion. Come now, let the octagon be 7, measured by feet along each side. This natural quantity, that is 8, if you multiply it by itself, you will produce; which if you multiply through, you will unfold. From these, if you subtract four times the sum of one side, no more than the remainders are left. If the half of these is taken, the area of this octagon is made known.

On the enneagon, a heading.

But let the enneagon be circumscribed along each side, which, namely, you will make 6, if according to the rule of our institution stated above you multiply it by itself; which multiplied seven times will produce the sum 109. If from these you subtract five times the quantity of one side, you will restore; if the half of these has been taken, it shows that the area of this enneagon is contained in half-feet.

On the decagon, a heading.

It remains for us to speak of the area-measure of the decagon. So let the decagon be described, bounded on its sides by the number ten. If the quantity of one side, multiplied by itself according to the already often-stated institution of our precept, has grown, it will produce [a product]. These, however, multiplied eight times, bring forth [a result]. But if you take the half of these, you will assuredly note that the area of this decagon is contained in feet.

If you do the same for the hendecagon and the remaining descriptions of many-sided figures, you will not stumble on any obstacle of error, in such a way that you proceed in the natural order in multiplying the sum of one side, and in this quantity which is born from this lateral multiplication, to be increased naturally, and the same lateral [quantity] to be subtracted naturally, and by such reasoning you find the area from the halves.

On the circle, a heading.

But since we have sufficiently disputed concerning angular figures for the studious reader, it remains for us to explain briefly concerning the circumduction of a sphere or circle. So let a circle be set down, designated in its circumduction by feet, and let the diameter be described by extensions of 14 feet. If its [square] has grown by itself, [numbers] will be born; multiplying these by 11 you will effect [a product], a fourteenth part of which, that is 154, reveals the area of this circle, as can be seen below.

There is another method of finding the area-measure of this circle: let the half of the circumferential quantity be taken, or the quantity that is the half, and let it be multiplied by the half of the diameter, that is by 7, and what comes from this multiplication reveals the area.

On the heading of the semicircle.

Since we have judged that these brief rudiments concerning circular theorems should be spoken of, we shall proceed at once to speak of the semicircle. Come, let a semicircle be described on its base, and having in its radius feet; if the area-measure of the area is unknown, it will be investigated by such a reckoning. Let therefore the sum of the base be multiplied by the sum of the radius, and this sum comes to ten times, producing; of which, a fourteenth part having been taken, that is 312, the surface of the area is completed, as is evident.

Let these things said at present concerning the area-measures of plane figures suffice; it remains that we treat somewhat more briefly, by reckoning, of a mountainous [figure]. Let even a mountain be inscribed, measured in feet in the circuit of its summit, and extended in feet from its foot all the way to the summit; but let the foot of the same mountain be marked in its circuit by a thousand feet. Let it now be proposed to be inquired how many acres are had in this mountain. Since this must be begun with such a reckoning, let those two circuits of the foot and the peak be joined, that is 1300; if the ascent, that is 800, is multiplied through the middle, it will be proved that the space of this mountain has 520 feet.

If, then, you divide this total by 28,800, you will confirm that it lies in this mountain, with only one thousand six hundred feet remaining.

But if the mountain has a circumference at its base of 1,600 feet, a circumference at its summit of one hundred, and a height of [mascensuoCO] feet, the acreage is to be found in this manner: let the sums of the three above-mentioned circumferences be joined, and from these is produced a third part, that is 1,400; multiplying the ascent of the mountain, that is 500, by this, they come forth: dividing these by the acre, you will effect no more than two hundred remaining units.

But if the mountain is crooked, that is uneven, with a circumference at its base of 1,400 feet, a slope at its summit of 200, an ascent on the right side of 850, and a height on the left flank of [lacuna] feet, the acreage of the level site is to be investigated in this manner. Let the half of the two circumferences gathered into one be taken, that is 800, and the middle part of the combined ascents, that is 800, and these halves multiplied by themselves produce the square measure, that is, of the above-mentioned mountain. From the foot-measure, moreover, you will easily find the acreage total according to what was said above.

Therefore, since we have sufficiently discussed, briefly and clearly, the principles of all the speculations to be included in this art, it remains for us to speak, as we promised, of the uncial and digital measure, and of points and minutes and the other small fractions, about to set forth that wondrous figure, necessary to this art and to the other disciplines of mathematics, which we learned under the guidance of Archytas.

The ancient geometers, then, most subtle investigators, and especially the Pythagoreans, when, dividing all things by fixed ratios of measures, they came even to those things which nature refused to have divided and cut, by the foresight of genius divided those things which were naturally indivisible, giving them fitting names. But when they had distributed fields by actus, by perticae—that is, by rods—by paces, by steps, by cubits, by feet, by half-feet, and by palms, not having a palm by which to divide that which was less than a palm but greater than a finger, they preferred to call it an uncia; and in the second place they set down the finger, in the third the stater, that is, the half-uncia, in the fourth the quadrant, in the fifth the drachma, in the sixth the scruple, in the seventh the obol, in the eighth the half-obol, which the Greeks call the cerates, in the ninth the siliqua, and they placed the point, the minute, and the moment by name. Having therefore devised these small fractions and published their names, they assigned to them multiform notations, which, because they were partly Greek and partly barbarous, did not seem to us to be joined to Latin speech.

Wherefore, not wishing to wrap an obscure matter in obscure and unknown signs of notations, in place of those same notations I set down the letters of the Latin alphabet in order, so that A corresponds to the uncia, B to the digit, C to the statera, D to the quadrans, E to the drachma, F to the scrupulum, G to the dobulus, H to the semiobulus, I to the siliqua, K to the punctum, L to the minutum, M to the momentum. Let there be described, therefore, with these letters, the figure of the unciae of which we have spoken, in this manner.

THE BOOK OF BOETHIUS ON GEOMETRY.

Geometry is the discipline of unchangeable magnitude and the contemplative description of forms, through which the boundaries of each thing are customarily declared, and it is also the visible doctrine of the philosophers, which in Latin is called the measurement of the earth. For since Egypt is said to have been first divided by the various forms of this discipline, on account of the need for boundaries of the land which the river Nile poured over in time of inundation, the masters of this discipline were formerly called surveyors. But Varro, the most learned of the Latins, records that the origin of this name came about thus, saying: First, indeed, the measurements of lands, with boundaries set, provided the useful things of peace to wandering and discordant peoples. Then the whole circuit of the year was divided by the number of months. Then even the months themselves were so called because they measure the year. Then he reports that the measurement of the circle of the earth was gathered by probable reasoning, and for this reason it came about that the discipline itself received the name of geometry, which would endure through long ages.

On the Usefulness of Geometry, a heading.

The usefulness of geometry is threefold: with respect to practical skill, with respect to health, and with respect to the soul. With respect to practical skill, as for mechanics and architects. With respect to health, as for physicians. With respect to the soul, as for philosophers. If we pursue this art closely and with diligent care and a measured mind, what is made manifest by the divisions stated above illuminates our senses with great clarity, and [enables us] to ascend in mind to that realm above, such as the heaven is, and to discuss that whole celestial mechanism otherwise by an investigable reasoning, and by the inspecting sublimity of the mind to gather from some part and recognize the maker of the world who veiled such great and such arcane things. For the world itself is borne along, gathered into a spherical roundness, so that it might enclose the diverse forms of things by the circuit of its own circumference, whence Seneca formed a book in an argument consonant with the philosophers, whose title is On the Form of the World. For in geometry we surely confess that a part is useful to be grasped and to be put into practice in actions; we judge that it is of benefit in all things, nor without cause did the greatest men also devote costly effort to this science, since geometry is divided into numbers and the forms of numbers. It is known to be necessary not only to the orator but to anyone educated at least in his first letters, because it stands firm in subtlety for the most delicate matters, is most useful for knowledge, and is very pleasant for practice. In causes, indeed, it is most frequently sought, because order is first of all necessary to geometry.

Is it not also [useful] for eloquence, since geometry proves subsequent things from prior ones, and uncertain things from certain ones? For this reason you will find many who enter this art in the manner of dialecticians and rhetoricians alike. For it uses syllogism in the dialectical manner, if the matter demands it, and those who are most powerful in grammar, which in Greek are called apodeixeis, prove the same, and certainly by enthymeme, which is the rhetorical syllogism, which in Latin is interpreted as "conception of the mind," which the writers on art are accustomed to call imperfect, and finally it proves of what form the circuit is, by how many straight lines it is contained. In what ways it is bounded, what that running-around line is if it makes a circle, which is the greatest and most perfect form among plane figures, in which it encompasses so many spaces. And if a square in equal hours makes squares again, a triangle triangles, itself more with equal sides than with unequal, and perhaps other more obscure things, which also the experiment of the work must follow. These things in planes. Likewise in mountains and hills it is clear even to the unskilled, because it is comprehended by the course of the sun alone and the motion of shadows, and is separated by the divisions of waters. Now let us come to the letter of Julius Caesar, which pertains to the origin of this art, so that neither the glory of the author himself may perish, so that the truth of the matter may come to our knowledge most fully; yet whoever wishes to read this letter studiously, introduced by certain compendia, will perceive the sayings of the ancients more clearly in brief.

Heading.

The deified Julius Caesar, a most vigorous man and conqueror of many peoples, trained his soldiers by the frequency of war, and for the sake of increasing his power entered the cities of illustrious men through the labors of greater wars; he received peoples and nations begging [for protection], he slew tyrants with the sword, and after he had obtained hostile land, having destroyed the cities of the enemy, he again established new cities, and, once more giving them the name of a colony, he enlarged their citizens.

He made soldiers into colonists, some in Italy, others in certain provinces. These cities which the deified Augustus, having been assigned [them], ordered for the army of the provinces on account of sudden clashes of war—[he ordered] not only that these states be finally encircled with walls, but also that rough and rugged places be sufficiently fortified, so that that place is especially a bulwark, both by the nature of the site and by the new dedication of fields; and he called the settlements of cultivation colonies, which colonies were assigned to those victors who had taken up arms for the sake of the occasion.

Therefore, so that we may not seem to have passed over anything, but rather to follow their examples, it will often be necessary to look to the model; and since the reasoning showed that the heights of mountains take precedence, by the kindled line of sight from a very lofty peak we have laboriously left markers built of stones, and there is a fortified and distinct quantity of places which persists separately through the divergences of waters on either side, a very well-known division, and other places of the banks' courses are preserved, and accordingly even if an enemy had wished to attack us, we could have broken them by assault from the nearest bank. For around the maritime region we judged the straight boundaries from compacted stones; we shall demonstrate the whole straight course of the boundaries, because all the columns which are placed at the sea are terminated by the shore of the sea. We have arranged the fields of the valleys by ordinary right, which we have called intercisive. But on the plain we have straightened the boundaries with the knife.

Very often there are many fields assigned, whose measure of boundaries, although it is diverse, nevertheless also differ from one another by 100 feet, by 150 feet, by 240 feet, by 300 feet, by 360 feet, by 420 feet, by 480 feet, by 600 feet, by 700 feet, by 840 feet, by 960 feet, by 1020 feet, by 10200 feet, by 10340 feet, by 10600 feet, by 10700 feet, by 10800 feet, by 20000 feet, yet in a long view with which we drive the boundary by markers.

If there is a thick boundary-mark of a narrow strip, and on the other side a long thick doubled one, these two outermost limits are called the decimanus and the cardo. Very many thousands of feet come together, and unless they end at the Alps they divide the fields on the right and left by straight lines containing one another.

You must make every measurement of this cultivation longer or broader in the middle, because what is longer in breadth is a bench, but what is longer in length is a strip.

There are estates of the well-deserving, and they have been assigned narrower in proportion to the estimation of their fertility.

Lean and dry places have been enclosed by a wider boundary. There are somewhat dry places which do not belong to the ordinary law, but if it has been agreed among the possessors, let them possess them; if it has not been agreed, it remains in the power.

Other more perfect places belong to the public law; just as many, if it is agreed among the possessors, they possess.

But there are public places which are written as forest and public pastures of the Augustans which cannot be alienated in that way, and they possess the guardianship either of public temples or of baths, which are called hills.

A field is excluded when it lies between the neighboring line and the centuries; it is called "excluded" because it is enclosed beyond the neighboring boundaries.

On controversies, rubric.

The materials of controversies are two, boundary and place, and of these one comprises whatever is disputed concerning a field. But since controversies marked out by these parts also have diverse conditions, as I have been able to comprehend, they must be named specifically.

The kinds of controversies are these: concerning the position of boundary markers; concerning the region; concerning the boundary; concerning the place; concerning the ownership of a house; concerning possession; concerning alluvion; concerning the law of a territory; concerning lands subject to cutting; concerning public places; concerning abandoned and excluded places; concerning sacred and religious places; concerning the inflow of rainwater, and concerning roads.

A controversy is between two or more neighbors. Between two, whether it lies in a straight line with respect to the others; between more, it makes a three-way boundary, and in some places even a four-way boundary, according to the nearest possessions; while they do not know this, they do not agree, and the possessors themselves create diverse controversies among themselves. Some litigate about the place, others about the line of the boundary, others pay attention to the estates. But in a greedy manner the origin of the case must first be sought. By nature, controversies of this kind arise through inheritances of opinion, wherefore it is litigated by ordinary law. First, however, in the judgment the question concerning possession should be settled, and then the land surveyor should be ordered to go to the places, so that with the truth made clear the dispute of this kind may be terminated.

The materials of controversies from a river are these: what is done by occupying fields does not have a claim for recovery, but what the force of the water has carried away does not; and for this reason, whoever it is that fortifies his own bank must fortify the bank without loss to another. But if the torrent of a river at some time runs so violently that it changes its channel, it occupies the fields of many across the bank, and often it even makes islands. But Cassius Longinus, a most prudent authority and judge of the law, established this: that whatever the water has carried off by licking away, the possessor may admit, since of course he ought to guard his own bank without loss to another. If, however, a greater force has run down and changes its channel into the boundaries of another, and an island forms in what it has run over, each person ought to recognize the measure of the greater river, and he whose land it pre-occupied in season will claim that island for himself, since it appears to have been seized not by the negligence of the possessor but by the violence of the storm.

A dry-ish field is recorded according to its own determinations; we enter it in its boundaries in the archive of Caesar, and whatever things have been granted or assigned to a colony by beneficium, whether in the vicinity or among other cities, we ascribe in the books of beneficia, and whatever else pertains to the equipment of the surveyors to the soil of the colony, but it ought to have it subscribed by the hand of the founder in the archive of Caesar.

A field of similar dry-ish condition is one excluded and not assigned, which, if it belongs to the commonwealth, to the Roman people, or to the colony itself whose boundary surrounds it, or to the Roman people, has not been given, remains in the power of him who was able to assign it.

The marks of the final boundaries in diverse regions, whether names, villages, or possessions, are these: testimonies of the fields to be divided between both possessors.

On mountains, in place of dry and rugged ground, we find marked stones.

The boundaries of the mountains are marked by round stones, that is, rounded ones, some marked with letters in the likeness of a column; the final boundary stones in the parts are heaps, that is, a pile of stones; trees left untouched by iron; a heap of rubble, that is, where stones gathered from both sides make a boundary line; likewise sacrificial stones as altars, in which places trees are seen to stand untouched, in which place the ancients, wandering, used to make sacrifice. In another place military roads make the end, which are fortified with a boundary stone. But another on the right of the mountains, that is, along the side of the mountain, running back, makes the boundary. Sometimes they make a burial place, and therefore tombs are to be followed which, at the outermost boundaries meeting together, await more junctions of fields. For all monuments bear witness to their owners.

There are boundary stones of the cursus established in the likeness of an inscription, fixed places of a stream, final cradles? or a stepmother, which is constructed with rules, scorpions where two wedge-shaped boundaries join together. If perchance in flat places where fields are established on the plain, they are found assigned in jugera. Likewise among the devoured rocks with trees left untouched, as I said above, sacrificial, then of the earth established in the likeness of a boundary line, millstones, pits, or boundary markers, lakes and legonatus, and calaviones constructed with structures. For sometimes square stones and writings, which indicate whose field, whose house, how much space they protect.

For not every boundary marker has been furnished with inscriptions, since in some places there are no inscribed stones, but stones set up in the likeness of boundary markers, which we call "itinerary" stones. For the mountains themselves wholly determine the places, but the boundary markers do not contain a single measure among themselves, by the command of Augustus Caesar, with Balbus the surveyor, who distinguished and declared the measures of all the provinces, and by the testimonies written above the boundaries of places are terminated.

For there are boundary markers to which no credence should be given; these are called "itinerary" markers. For all boundary lines must serve the public road, which divide private lands on the right and on the left, and in the middle is the public road. These are not in all places, yet under all boundary markers a sign ought to be found; therefore whatever is found in place of a boundary marker ought to be observed and guarded, so that it is directed from one to one, and if by night or by day, to the mark.

For thus are the customs and observances of certain law, that a sign is always placed on all boundary markers, or we find some ashes, or coals, or potsherd, or bones, or glass, or a mass of iron, or bronze, or chalk, or gypsum, or a clay vessel, which they also trampled down with certain fragments of stones and confirmed with diligent care, so that it might stand more firmly. Such things, then, they made as a sign between the owners, between whom the boundaries were being marked out.

But boundaries are not present in all places; rather, there are many other countless testimonies. Read happily and take care to understand; one who understands what he sees can remove the intent and dispute over fields. Yet those surveyors who are going to judge, and those whom they call in as guarantors, ought to have prudence. In judging, a good and just surveyor ought to act so that he is moved by no ambition or sordid gain, and he ought to preserve the reputation of his measures and morals. For by every craftsman the truth must be guarded; those who set false things in place of true are excluded. Some sin through imprudence, some through ignorance. Therefore, changing in the profession the things that are generally added as true, through controversy by argument and by conjecture, craftsmen are compelled to measure even superfluously, but it will befit a man who is to judge as a craftsman to keep this safe.

The names of the surveyors, under a heading.

Of Hyginus, of Marcus, by order of Caesar Nero. — Of Julius Frontinus, of Junius, by order of Claudius Caesar. — Of Siculus. — Of Flaccus, of Nipsus, of Hyginus, of Euclid, of Cassius. — Of Agenus, of Balbus, by order of Tiberius Caesar. — Of Urbicus, of Mensor Longus. — By order of the Emperor Severus and Antoninus. — By order of the Emperor Vespasian. — By order of the Emperor Hadrian. — By order of the Emperor Trajan. — By order of the Emperor Augustus Caesar. — Of the Emperor Nero. — By order of the Emperor Valentinian. — By order of the Emperor Theodosius. — By order of the Emperor Arcadius. — By order of the Emperor Honorius. — By order of the Emperor Constantine.

The names of the boundary stones and the positions of the chests: the rubric.

The orthogonius sends a right angle straight. The isopleurus, set directly beneath. — The isosceles. The boundary line.

— The exculerms, or hexagineus. The running spatula. — The escuiteiat.uslatenbus. The boundary placed in reverse.

— The sumbus, or trapezideus. Likewise the running spatula. — The isosceles. The quadrifinius. — The lone triangle boasts other things. Likewise the quadrifinius. — The parallelogram, the pentagon. The gamatus boundary. The hexagineus. The lineated boundary, that is the quadrifinius. — The septagenus. Likewise the quadrifinius. — The sinagonus. The noverca. — The boundary written with a Greek letter. The simmatus. — The boundary sharp at the top. The centustatus. — The circular pyramus, likewise similar to a sharp one. The trivortinus. — Likewise the pyramus similar to a life cut short. The amicirculus — The complete rhombus, the amHygmeus. The varoberinus. — The squared amicirculus. The thrusting one. — Thrice less, the agusteus. The boundary sharp at the top. — The running boundary. The millstone. — The boundary of the boundary. The monument. — The burial with bones at the boundary. The mauspteus. — The boundary among the small bricks. The final chest. — The quadrifinius boundary. The lippus. The round boundary. The kalasiones. — The boundary beneath which lies an angle. The quadrifinius boundary.

You who wish to be a perfect geometer, cover all these things which have been summarized below. For first of all one must know the art of arithmetic, which contains the causes and divisions of numbers, that is: what is definition and division concerning even and odd numbers; — what is a composite number, and what an incomposite; — what is a perfect number, and what an imperfect; — what is a divisible number, and what an indivisible; — what is a particular number, and what a superpartient; — what is a superfluous number, and what a diminutive; — what is a multiple number, and what a submultiple; — what is a solid number, and what a spherical; — how geometry was discovered; — what geometry is; — by what?

utility; what is the order of prescription; what is the principle of pre-position; what of disposition; what of distribution; what of description; what of demonstration; what of conclusion; of what sort is a straight line; of what sort is the surface of a line; of what sort is the dividing line; how many kinds of extremities there are; how many kinds of summits; how many kinds of angles; of what sort is a plane angle; of what sort is an obtuse angle; of what sort is a blunt angle; of what sort is a right angle; of what sort is an acute angle; of what sort is a divided measure; how much a stade extends; what is a needle; what are the climata; what is a centuria; what is a league; what is an arapennis; what is a jugerum; what is a centuria; what is a point; what is a diameter; what is a parallelogram; what is a figure; what is a circle; into how many parts the division is made.

If you know all these things to fullness, you know the separation of places. For those who are ignorant of the rule of this art set forth many false things in place of true ones.

D. How was geometry discovered?

The Egyptians say that geometry was discovered because of the need for boundaries of land, which the Nile used to flood over in time of inundation.

D. From what is it called geometry?

Geometry is named from the measuring of the earth, by which the boundaries of each piece of land are customarily marked out.

D. What is geometry?

M. Geometry is the discipline of magnitude and figure, which is contemplated according to magnitude.

D. What is its intent?

M. The intent of Euclid is twofold, looking to the student and to the natures of things. It looks to the student, because he ought to begin from these introductory matters for the sake of ease, for the sake of brevity, and because in the questions there is on this account no difficulty. For the nature of things, because the physical science and the doctrine of the Timaeus, that is, of Plato, are known to demonstrate very many things by geometry.

D. What is its utility?

M. The utility of geometry (as we have said above) is threefold: for capability, for health, for the soul: for capability, as for mechanics; for health, as for physicians; for the soul, as for philosophers.

D. What is the order of geometry among the disciplines?

M. In some respects it is second after arithmetic, in some respects third.

D. How is the title inscription to be understood?

M. For the title is the preface of the elements, which are the simpler figures, and from these others are composed which are also resolved into these.

D. If the book is more properly so called?

M. This book is said to be according to the arrangement of Euclid; according to demonstration or discovery, it is generally said to be of others.

D. Into how many parts is it divided?

M. This book is divided into four parts: into planes, into arithmetical matters, into rational and irrational lines, and into solids.

D. What are the steps in geometrical demonstration?

M. Proposition, disposition, distribution, description, demonstration, and conclusion.

There remains for us to hand down a certain most profound discipline, which by reason pertains especially to the integrity of all nature and of things. For great indeed is the fruit in this science, if one is not ignorant that defined goodness, falling under science and always imitable and perceptible by the mind, is the first nature, and the perpetual beauty of its substance, while the infinite disgrace of malice, resting on no proper principles of its own, but always wandering by nature, is composed away from every definition of principle as if by some mark of the best impressed upon a figure, and is held back from that flood of error. For.

For the mind, strengthened by pure intelligence, restrains the excessive and unbridled desire of the soul, like a kind of ruler. Nevertheless, the matters that were discussed at greater length by Nicomachus concerning numbers, or shown by Varro concerning measures, we have gathered with moderate brevity. And the things that, having been passed over too quickly, offered a narrower approach to understanding, we have opened up with a moderate addition, so that at some point we might use our own formulas and descriptions for the clarity of things—something that cost us how much vigilance and sweat the sober reader easily recognizes. And these forms of inequality, so to speak, we have gathered by tempering them with a balanced goodness; the reader himself will approve the things we shall draw from the wealth of Greek letters into the treasury of Roman speech. If any things have shone forth from the teachings of wisdom, let them be approved through us by the judgment of the wisest. You see, therefore, that the outcome of so great a labor awaits your examination alone, as reader, and does not come forth into public ears unless it is supported by the assent of learned judgment—which should seem in no way surprising, since that work which pursues the discoveries of wisdom relies not on the author's own judgment but on another's. For wisdom is the causes and divisions of numbers, and the true knowledge and complete comprehension of these things; whoever despises this, that is, the path of wisdom, to him I declare that he must not philosophize rightly.

This is arithmetic; for it is prior to all things, not only because God, the founder of this worldly mass, held it as the first model of his own reasoning, and established all things in accordance with it—whatever things, fashioned by reason, have found concord through the number of an assigned order—but arithmetic is also shown to be prior for this reason, that whatever things are prior by nature, when these are removed, the things that are posterior are at the same time removed. But if the posterior things perish, nothing is changed from the state of the prior substance, as an animal is prior to a human being. For if you remove the animal, at once the nature of the human being is also destroyed. If you remove the human being, the animal will not perish. Yet properly the nature of numbers itself has preceded all things. All things whatsoever that have been constructed by the primal nature of things appear to have been formed by the reasoning of numbers. For this was the principal model in the mind of the founder; from this the multitude of the four elements was borrowed, from this the alternations of times, from this the motion of the stars and the revolution of the heaven. Yet properly the nature of numbers itself has established all the courses of the stars and every astronomical reckoning. For in this way we gather risings and settings, in this way we observe the slownesses and speeds of the wandering stars, in this way we recognize eclipses and the manifold variations of the moon, because—since, as has become clear, the use of arithmetic is prior—let us take from here the beginning of our discussion; this same thing seems to occur in geometry or in arithmetic.

For if you take away number, how can we comprehend the triangle or the square, or whatever is treated in geometry, all of which are named from numbers? This will be clear if we understand that all inequalities arose from first principles, so that equality itself, possessing as it were the force of a mother and root, pervades all the species and orders of inequality. For let us take three twos, or three threes, or three fours, or however many beyond these one wishes to posit. For what happens in these three terms happens likewise in the rest. From these, therefore, according to the order of our instruction, you may see that multiples are first born if they are converted, and among these doubles first, then triples, then quadruples, and those following in the same order. Again, if the multiples are converted, superparticulars will arise from them: from doubles, sesquialters; from triples, sesquitertians; from quadruples, sesquiquartans, and the rest in this manner. But from superparticulars, when converted, superpartients must be born, so that from a sesquialter is born a superbipartient, a sesquitertian begets a supertripartient, as from a sesquiquartan a superquadripartient. But when the former superparticulars are placed directly and not converted, multiplex superparticulars arise.

But from straight superpartient ratios, multiple superpartient ratios are produced; and these are the three rules: that you make the first number equal to the first, the second to the first, and the third to the second, and the first to the two seconds and the third. For when you have done this in equal terms, those which are born from them will be double. And if you do the same with these doubles, triples are produced, and from these quadruples, and thus to infinity you will unfold all the forms of multiple number. Those indeed whose parts extend beyond what is sufficient are called superfluous, as are 12, or 24. For these, compared with their parts, obtain a greater sum of parts than the whole body: for the half of twelve is 6, the third part 4, the fourth part 3, the sixth part 2, the twelfth part 1, and all this heap runs over into 16, and they surpass the multitude of their whole body. Again, the half of the number is 12, the third 8, the fourth 6, the sixth 4, the eighth 3, the twelfth 2, the twenty-fourth 1, which all repay, in which matter it is manifest that the sum of the parts is greater, and overflows beyond its own body. And here indeed that whose parts are composed is surpassed by the multitude of the whole term, as 8, or 14; for the number eight has a middle part, that is 4, it has also a fourth, which is 2, it has also an eighth, 1, which all reduced into one gather seven, namely concluding a sum smaller than the whole body.

Again, fourteen has a half, that is 7; it has a seventh, that is 2; it has a fourteenth, that is 1. If these are gathered into one, the sum of the number ten grows up, that is, less than the whole term. But those which are in this way, like that former one whose parts surpass it, seem such as if someone were born with many hands beyond nature, or joined with a double body, or whatever monstrous thing of nature has crept in through the multiplication of parts.

But those lesser ones, as if by some necessary part naturally taken away, one were born with less than an eye, or cut short in another limb, would obtain the loss of its whole fullness by nature; but between these, as between equals, the intemperance of the middle has obtained the mean limit—that number which is said to be perfect, namely the emulator of virtue, which is neither directed by superfluous increase nor relaxed again by contracted decrease; but obtains the middle, which by its equal parts neither swells with abundance nor lacks in want, as for instance 28. For the six has a middle part, that is 3, and a third, that is 2, and a sixth, that is 1; if these are reduced into one sum, that is 3, 2, 1, it is found equal to the whole body of the number by its parts; but it has a half, 14, and a fourth, 7, and a seventh, 4, and a fourteenth, 2, and a twenty-eighth, 1; these reduced into one will equal the whole body by its parts, for joined in one the parts make it up.

But in these matters too there is a great likeness of virtue and vice; for you will rarely find perfect numbers, and those easily countable, inasmuch as they are few and brought forth in a most constant order. But you will find superfluous ones infinite, and disposed in no order at all, but here and there and in disorder, and begotten by no certain limit. Now the perfect numbers are: within the number ten, 6; within the hundred, 28; within the thousand, 416; within ten thousand, 128. And always these numbers end in two even [digits], both 6 and 8, and always by alternation the ends of the sums come to these numbers. For first 6, then 28, after these 416. The same six, which is first, [comes] after 128 also, the same eight which is second.

- But of the greater inequality there are five parts: for one is called multiple, another superparticular, the third superpartient, the fourth multiple superparticular, the fifth multiple superpartient. To these five parts of the greater [inequality], then, are opposed other five parts, which are named by the same names, one by one, for the species of the lesser, differing only by the prefix alone. For it is called submultiple, subsuperparticular, subsuperpartient, submultiple superparticular, submultiple superpartient. In the first part, if the number is multiplied, it is called multiple. In the second part it is called superparticular. In the third, superpartient, that is, the fourth, multiple superparticular. In the fifth, multiple superpartient.

But of the lesser number, one subsisting as some part and the same through its parts is extended according to the pattern and multitude of the greater numbers.

On even and odd numbers, a heading.

Now the description that is set down below was made in this way: however many times the number three multiplied in the order of the equally even numbers, whatever were produced from it were arranged in the first row. Again, those that were born from the same multiplier five were set in the second place. Then those that the seven, multiplying the rest, produced, we set in the third place, and we complete the same in the remaining part of the description. Therefore the rationale of the description arranged above is this: if you look to the width, where there is one mean of two terms, and you join the terms themselves, you will find them double their own mean, and they make 56, whose mean is 28; and if you join them, they make 150, of which the middle term among them is found to be the mean. But indeed where they have two means, both extremes joined together become equal to both means, as 12 and when you have joined them they are made; if you apply their means to one another, that is, and they remain. And in another part of the width the numbers that are made are noted in the same order, nor did the rationale of either width differ in any respect. And you will note the same in the remaining numbers in the same order, and this according to the form of the equally odd number, in which this property exists, as has already been said above.

Again, if you look to the length, where two terms have one middle, what is produced from the multiplied extremes is produced if the middle term takes the increases of its own plurality. For twelve times twelve makes 576; and their middle term, that is 24, if it is multiplied, will again produce the same. And again if it is multiplied into 96, they make 2304, whose middle term, that is 48, if it is multiplied by itself, the same is produced. But where two terms enclose two middles, what is produced by the multiplied extremes is the same as is restored to either sum by the multiplied middles. For by twelve multiplied into 192 they are produced, and their two middles, that is 24 and 48, if they are multiplied by themselves, will restore the same into 576, and this is by the imitation and kinship of the even-times-even number, from which, by participation drawn, this property is known to be inborn in it. And on the other side of the length, the same reckoning and description is noted, whereby it is manifest that the whole number is produced from the two above. which it retains the properties of. But since by nature and according to the proper consequence of order we have placed the multiple species of inequality before all others, and have shown that it is the first species—although this becomes clear to us by the order of the later work—touching on it here too, let us teach most plainly and briefly what we have proposed.

For let there be such a description in which the natural order of continuous numbers is set in order up to the number ten, and in the second row let the double order be woven, in the third the triple, in the fourth the quadruple, and so on up to the tenfold. For thus we shall recognize how the species of the multiple will be first with respect to the superparticular, the superpartient, and all the others, and we shall at the same time inspect certain other things, both most subtle for precision and most useful for knowledge and exceedingly pleasant for practice.

If, therefore, the first two sides of the proposed figure, which form an angle from one to ten, and the ten proceeding onward are looked at, and the rows beneath these are compared—those namely which, beginning from the angle, place their limit at twenty—the double, that is the first species of multiplicity, is shown, such that the first surpasses the first by unity alone, that is two surpasses one, the second exceeds the second by two, as the four exceeds the two, the third exceeds the third by three, as the six exceeds the three, the fourth transcends the fourth by the number four, as eight transcends four, and by the same sequence all pass beyond the lesser by its plurality.

But if the third angle is examined, which begins from nine and extends its length and breadth by thirty numbers on either side, and if this is compared with the first length and breadth, a threefold species of multiplicity occurs, so that this comparison is made through the tenth letter, and these numbers will surpass one another according to the naturally made connection of parallelism; for the first surpasses the first by two, as three surpasses one; the second surpasses the second by four, as six surpasses two; the third surpasses the third by six, as nine surpasses three; and the rest increase by the same manner of progression. This, indeed, natural integrity itself has set before us, with nothing contrived by us beyond it, as appears in that very model of description.

But if anyone wishes to compare the limit of the fourth angle, which is marked by the quantity of the number sixteen, and its length, which terminates in forties, with the foregoing, compared proportionally through the form of the letter T, he will note the multitude as fourfold, and for these there is an orderly progression over themselves, and that the first surpasses the first by three, as four surpasses unity; the second conquers the second by six, as eight surpasses two; the third passes beyond the third by nine, as twelve surpasses three; and the following ones always pass beyond by the amount of three added to the sum. And if anyone looks at the lower angles, the same will come about through all the species of multiplicity up to the tenfold by a most orderly arrangement.

But if anyone seeks superparticular ratios in this arrangement, he will find them in the following way. For if he marks the second angle, whose beginning is the number four, and over it lies the number two, and if he applies the following row to this, the sesquialter proportion is declared. For the third term of the second row is sesquialter to the second, as three to two, or six to four, or nine to six, or twelve to eight; and likewise in the rest that are in the same series of numbers, such a pairing is mixed, and no dissimilarity of variation will creep in; yet the same excess of the sums is in this case also as was in the doubles. For the first exceeds the first, that is, three exceeds two, by one; the second exceeds the second by two; the third exceeds the third by three; and so on. But if the fourth row is compared with the third, as four to three, and you follow the rest in the same order, the sesquitertian comparison is gathered, as four to three, or eight to six, or twelve to nine. Do you see how in all these the sesquitertian ratio is preserved? Moreover, if, doing the same, you compare the rows that follow with those beneath them, you will recognize all the species of superparticular ratios without any hindrance. But this is divine in this arrangement, that all the angular numbers are tetragons. Now a tetragon is called, to speak most briefly (which will also be explained more fully), that which two equal numbers multiply, as is also in this arrangement: for one times one is one, and it is a tetragon in power.

Likewise, twice two are, and three times three are 9, which multiplications have completed in their beginnings into themselves. But those that are around them, or is around the angles, are long-sided numbers; for I call "long-sided" those which numbers multiplying one exceeding the other produce; for around two are six, but two are born from one and two, when you have multiplied one twice. But unity is preceded by the binary by unity, indeed from two and three, for twice three yields six. But six and twelve enclose the nine, which twelve are born from three and 4. For three times 4 make 12, but the six from two and three, for twice three make 6, all of which are produced by sides greater by one. For when they are born from the binary and the ternary, they exceed the number two by one, and all others of the same kind are, so that in the first and second order the terms are multiplied by one another and produced, so that what is born from two long-sided numbers placed on either side, with twice the middle square, is a square. And again, what is born from two squares on either side and one middle long-sided number twice made, is itself also a square, and so that of the angles of the whole description placed at the corner squares, the first is unity, but the other which is opposite is the third; but to one of the angles on either side they have second unities, and the two angles of the corner squares make equal what is contained under them, to that which is made by one of those which are on either side of the angles.

For there are many other things in this diagram that can be usefully and admirably weighed, which for now, on account of the restrained brevity of the introduction, we allow to remain unknown; but now let us turn our purpose to what follows.

The end of the book of Geometry of Anitius Manlius Severinus Boethius. In which work, if anything further seems to be required by anyone, let him consult the good and the fair-minded, for we have emended very many passages; in certain ones it has seemed best to leave each person his own judgment, and to change nothing rashly. Certainly, as far as diligence, industry, and expense could accomplish, nothing has been omitted by us.