Book of Various Mathematical and Physical Speculations

Some propose that a number is to be divided into two such parts that the sum of the squares of the said parts is equal to another proposed possible number—"possible," I say, for if such a proposed number were less than the product of the whole of the first number by its half, such a result would be impossible. Wishing to carry this out, let us take the first proposed number, which we multiply by itself; from this square let us subtract the second proposed number, then let us double what remains; I then bid this double to be subtracted again from the same first square, and after the square root of the residue is taken and subtracted from the first number proposed, then half of the residue will be one of the two sought parts of the first number.

For example, let 20 be proposed to be divided into two such parts that the sum of their squares is equal to 272, which number is greater than 200—greater, I say, than half of the square, 400, of the 20 itself. Now let this number 272 be subtracted from the square 400, and 128 will remain; I bid this to be doubled, and 256 will indeed be produced; which likewise being subtracted from the total square, 144 will remain. I wish the root of this to be taken, which will be 12; and being subtracted from 20, the first given number, 8 will remain, half of which will be 4—one part of those sought—which shall be subtracted from the first proposed number 20, and 16 will remain for the other part.

A geometric diagram showing a square subdivided into several rectangles and squares, labeled with letters d, m, f, g, b, t, a, c, k. The letter 'e' is at the center of the subdivision. This figure illustrates the geometric proof for dividing a number into parts such that the sum of their squares equals a given value.

For the sake of demonstrating this, let us first consider the known square $a.c$ of the number $a.b$ first proposed, which is conceived as divided into two squares $d.e$ and $e.b$, and two supplements $a.e$ and $e.c$; moreover, the number of the sum of the two squares $d.e$ and $e.b$ is given as the second proposed [number]; from which, the sum of the two supplements $a.e$ and $e.c$ will consequently be known. When this has been doubled, and these four supplements arranged in thought, as appears in square $f.g$ (although the same would result if only the eighth [proposition] of the second [book] of Euclid were applied), it is equal to square $a.c$, so that with the four supplements of the known number conceived in square $f.g$, the number of the partial square $h.i$ will consequently be known, and also its root, which, when subtracted from number $a.b$ or $f.n$ (which is the same), the first proposed, there will be left the known number, the double of $x.n$ or $t.b$, one part of the whole $a.b$, by which this problem of mine will be true.

T H E O R E M X X V I I I.

If anyone should seek another method of performing this thing, let it be by this, after finding the number of this supplement, since it was stated in the preceding theorem by what method the double of the supplement itself is manifested.

In the figure below, let us consider line $a.b$ as the first proposed number, and let the product $a.e$ be equal to the supplement $a.e$ of the first preceding figure, and then let the procedure continue according to the order handed down by the ancients, with half of $a.b$ reduced to a square, namely $b.c$, which will be $b.d$; from which $a.e$ is then subtracted, wherefore there will remain

---

the known square $e.d$, the root of which will be equal to $c.e$, which, when joined to the half $c.a$, will give what was proposed, according to the fifth [proposition] of the second [book] of Euclid.

T H E O R E M X X I X.

What is the reason why, when twice the product of two numbers multiplied by each other is subtracted from the sum of their squares, the remainder is always the square of the difference of the two numbers?

For example, if two numbers 16 and 4 were proposed, twice their product would be 128; which being subtracted from the sum of their squares, namely from 272, there would remain 144, the square root of which would be 12, as the difference between 4 and 16.

That we may know this, let the two proposed numbers be signified by two lines, the greater $q.g$ and the lesser $g.p$, joined directly, upon which the total square $a.p$ is constructed, in which the diameter $a.p$ is conceived; and from point $g$ let the parallel $g.n.e$ be drawn, and from point $m$ the parallel $n.s.r$; from which two products $q.n$ and $n.u$ will be given, each equal to the product of $q.g$ into $g.p$, and $a.n$ and $n.p$ [will be] the two squares of the said proposed numbers, which is proved more than sufficiently by the fourth [proposition] of the second [book] of Euclid. Let us then conceive $n.o$ equal to $n.p$, and from point $o$ let $o.m.t$ be drawn parallel to $r.s$, and $o.e$ to $n.c$; wherefore from the things brought forward by Euclid in the eighth [proposition] of the second [book], the quantity $m.n$ will be given equal to $q.n$, the product of $q.g$ into $g.p$, and the quantity $o.c$ less than that product by the quantity of the square $n.p$; from which the quantity $m.n.e$ together with the square $n.p$ will be equal to twice the product of $q.g$ into $g.p$. But these two quantities are parts of the two said squares, and what remains, $m.e$, is the square of the difference of one proposed number from the other, as anyone may consider in the figure below. Therefore, this truth will be manifest.

A geometric diagram of a square with internal grid lines and a diagonal from top-left to bottom-right, labeled with letters a, e, u, m, n, o, r, s, p, q, i, g. It illustrates the geometric relationships described in Theorem XXIX.

T H E O R E M X X X.

Why do those who divide the greater of two proposed numbers by the lesser, if they multiply the result by the greater number, find the product equal to the result of the division of the square of the greater number by the lesser?

For example, if two numbers 20 and 4 are proposed, and 20 itself is divided by 4, it will give five; then 400, the square of 20, divided by the former 4, will give 100; which result is equal to the product of 20 and 5, the first result.

A geometric diagram showing two rectangular shapes of different sizes, with the larger labeled 'M' on the left. Points and lengths are marked with letters s, x, n, e, o. This diagram supports the mathematical proof of Theorem XXX.

For the sake of this speculation, let there be two numbers, which are signified by the lines $x.u$ and $x.s$, the greater and the lesser; then let number $u.x$ be divided by $s.x$, and let the result be $x.n$; afterward, let the square of $u.x$ be $x.o$, and let the product of $n.x$ into $u.x$ be $x.e$, which I say is equal to the result of the division of the square $o.x$ by $s.x$, which is $m$. For it is clear from the definition of division that there will be such a proportion of $u.x$ to $n.x$ as there is of $s.x$ to the unit, and as the square $o.x$ is to the rectangle $e.x$, so it will have