Division

Proposition 27

Such a division can be performed on all lines, on a line Expressible in length, on one Expressible only in square, on a Medial line, on a line of 1 of the 12 kinds we have listed, and on all other kinds of line

We are concerned only with two of these kinds [of line] which coincide with kinds already explained according to the two lines which are to be sectioned in this way.

For it [the line to be sectioned] is either Expressible in length or is a Mizon.

If the line which we propose to section is Expressible in length; the greater part of the sectioned line will be an Apotome of the fourth kind;

Corresponding to it there will be a Binomial of the same fourth kind, having the same terms as it has.^‘^ But beware of confusion, for this part is called greater in relation to the proposed line;

but the same part is here called an Apotome, not in relation to the proposed line; but qualitatively. If you want to know what it is an apotome of, the answer is that it is an apotome of some line which is commensurable with the proposed line only in square, which namely is the side of a square I times that of the proposed line.

Let GA be the line which is to be divided, and let it be Expressible in length.

Construct a right angle GAM and let AM be half the length of GA, and, having joined the points G and M, taking center M and radius GM let there be drawn the semicircular arc PGX, and let AM be produced to cut this arc at P and X, and let there be constructed on the line PA the square PO.

Therefore the line GA is divided in proportional section at the point O. So the line AO is the greater part of the line GA that has been divided in proportional section; but the same line AO or the line AP, which is equal to it, is an Apotome not of the line GA butof the line MP or MG, which when square is equal to the sum of the square of GA and of AM, half of GA: so if the square of the line GA were 4 the square of AM would be 1.

Thus the square of the line GM would be 5. Insofar as AO or AP is an Apotome it corresponds to the binomial AX: and their common terms are MX, or MP, or MG, and AM.^^

Now the fact that AP is an Apotome, and AX a Binomial, each of the fourth kind, is proved as follows. For both the terms MX and MA are expressible; however, they are commensurable only in square, because MX (that is MG) is the side of a square of area 5 units, [the units being] such that MA is the side of a unit square. And the ratio of 1 to 5 is not that of one square number to another. Then the difference of the square 1 and 5 is 4, a square number whose side, 2, is Expressible in length, and is equal to the proposed line GA. These are the marks of the terms of Binomials of the fourth kind, in the definitions preceding proposition 48, and of Apotomes, in the definitions preceding proposition 85 of Euclid.

Lastly, if the Expressible line GA is divided in proportional section, its greater part, OA, and the line compounded from OA and AG^"^ are both of the fifth degree of knowledge.

For if their squares are combined their sum is Expressible, namely three times that of the square of the expressible line GA by Euclid XIII.4.

Their Rectangle is also Expressible, because it is equal to the expressible square of the line GA, since GA is a mean proportional between the part OA and the compound of OA and AG, as was assumed.

Proposition 28

On the other hand, if any line Expressible in length is thus divided in proportional section, its smaller part will be an Apotome of the first kind.

So if the Expressible line is GA, as before, and when it is divided in proportional section its greater part is AO and its smaller OG; OG will also be an Apotome, by Euclid XIII.6.

Again note that OG is called an Apotome qualitatively, not in relation to the line GA, expressible in length, of which OG is the smaller part; nor in relation to MG, or MP, of which the line AO or AP is an Apotome; but GO has its particular Terms.

For since by Euclid X.97®” the square on any Apotome, and thus also the square PO, applied to an Expressible line (as here to GT equal to the line GA) produces as breadth GO, an Apotome of the first kind^^: on the other hand, the line AO was an Apotome of the fourth kind. So for the former, GO, the greater term is Expressible in length; and for the latter, AO, the greater term, MP, was expressible only in square.

On the other hand, because the terms are commensurable only in square; it is necessary that the Smaller term, or Prosharmozusa, of the line GO, should be expressible only in square, since for the line AO the smaller term, AM, was expressible in length: however, for both it is true that the difference of the squares of the [individual] terms is a square whose side is expressible in length.

What are the Terms of GO, a first Apotome, I leave for others to find.

As the line GO is a First Apotome, its Prosharmozusa is a single unique line, by Euclid X.79.*’^ This line must be such that its square shall be Expressible, but not by a square number; and the line must, together with GO, make a single line Expressible in length; and by X.30,^’^ if this one complete line is made the diameter of a circle, say PX

If the Prosharmozusa, somewhat longer than PA (provided the whole were equal to the line PX) were from one end of the Diameter, X, applied to the circumference^"^ to give the line XG; then the line joining the points G, P must be commensurable in length with the line PX.^^