COR. 2. Any aliquot part or submultiple of a common measure, is also a common measure.

COR. 3. By repeating the process with the remainder and lesser magnitude, and again with the new remainder (if there be one) and the preceding, and so on, the greatest common measure of two given commensurable magnitudes, A and B, may be found.

S. 1 Hyp.

Let 2 B = A-R; 3 R= B−R2 = R−R3

and 5 R, R, exactly.

2 Cor. 1, N, V. Then

the greatest com. meas. of A & B, is also the greatest for B & R;

and the greatest for B & R, the greatest for R & R and so on;

and

R is contained in itself and also in

R2 exactly;

.. R is the greatest common measure of R & R3,

and also the greatest common measure of A & B.

COR. 4. Any two commensurable lines are to one another as the numbers denoting the number of times that they respectively contain their common measure; thus, if the com. meas. of A B be EF, taken 5 times; and that of C D, the same E F, taken 7 times, then the ratio AB: CD the ratio 5: 7.

SCH. If, on continuing the process, we never arrive at a remainder which exactly measures the preceding remainder, the magnitudes are incommensurable.

USE & APPL. The Process for finding the greatest common measure of two numbers is included in this Prop. N, Bk. V:--- for, by dividing the greater number by the less, and finding the remainder ;--then, by dividing the less by the remainder, and finding the second remainder, if there be one; and by dividing the first remainder by the second, and finding the third remainder; and so on, until a remainder be found which exactly measures the last preceding remainder; this final remainder that exactly measures the last preceding will be the greatest common measure.

PROP. O.-THEOR.

The diagonal and side of a square are incommensurable.

CON. Pst. 3, I. A circle may be drawn at any distance from the centre. 11, I. To draw a perpendicular from a given point.

DEM. 32, I. The int. /s of every A are together equal to two rt. s.
20, I. Any two sides of a ▲ are together greater than the third side.
Def. 15, I. Every point in the Oce of a circle is at an equal distance
from the centre.

Cor. 3, 16, III. Tangents to a from the same are equal.
37, III. If from a point without a circle there be drawn two lines,
one of which cuts the circle and the other meets it; if the rect-
angle contained by the whole line which cuts the circle, and the
part of it without the circle, be equal to the square of the line
which meets it, the line which meets it shall touch the circle.
The s at the base of an isosc. A are equal.

5, I.

6, I.

E. 1 Hyp.

2 Concl.

If the s at the base of a triangle, are equal the sides opposite are equal.

Let A B C be the half of a square; A C the diagonal, and BA, BC two sides conterm. in B.

=

.. AC- AB AD, and AD is < AB. Also CD = CB = AB, and AD < AB; and ED and EB are tangts. from the same. E;

And. A D E is a▲, and ▲ A DE a rt. ; and A = art. 2;

and.. AD = DE = EB.

Now when AD the first rem., or its equal
E B is taken from A B,

then the rem. AE is the diag. of a sq., of
which A D, DE are the sides.

The same process as before will then have to be followed out;

and when AD as side has been taken from A E as diag.;

then the rem. lines will again be side and diag.

But a diag.
remainder.

a side always leaves a

.. in this process there will ever be a rem. ; .. the process will never terminate;

and .. AC the diag. of a sq., and CB a side, are incommensurable.

Q. E. D.

PROP. P.-THEOR.

If four straight lines, A, B, C, D, be proportionals, (whether commensurable, or incommensurable,) the rectangle under the extremes AD will be equal to the rectangle under the means B. C.

Book II. p. 145. The numerical area of a rectangle is obtained by supposing the two sides containing the rectangle to be divided into a number of linear units of the same kind, as inches, feet, &c., and then multiplying the units on one side by the units on the other: the product represents the area or enclosed space.

R

Cor. 1. Pr. 29, I. § 4. p. 18. Geom. Plane, Sol. & Spher. "If there be two st, lines, one of which is contained an exact number of times in one side of a rectangle, and the other an exact number of times in the side adjoining it; then, the rectangle under those two st. lines shall be contained as often in the given rectangle, as is denoted by the product of the two numbers which denote how often the lines themselves are contained in the two sides."

Def. 5, V. Criterion of the Equality of Ratios. Pr. M, Bk. V.

First.

C. 1 Assum.

2

3 Pst. 2, V.

D. 1 C. 3

Let A & B be commensurable, and :. also C & D.

Take any com. ratio whatever, as 7: 5;
and for com. measures, M and N;

let M be contained in A seven times,in B five times;
& N be contained in C seven times, in D five times.
... A= 7 m, and D = 5n,

.. rect. A D = 7×5 (m.n)

= 35 m. n.

2

3 Sim.

Second. Let A & B be incommensurable, and
.. C & D.

C. 1 Pr. M, V.

2

D. 1 Pr. M, V.

3 Pr. M, V. 4 Rec.

Find st. lines P & Q which
approach nearer A & C than
any assigned difference,
and let P&Q contain like
parts of B & D, so that

P.

D =

Q. B.

AMB CND

Now, by taking like parts of B & D, continually less and less, P & Q, increase towards A & C within any assigned difference;

.. P.D and Q. B, by increasing together,
approach more nearly A. D and C. B than
any assigned difference;

.. rect. A. Ꭰ =rect. C. B or B. C.
Therefore, If four st. lines, &c.,

Q. E. D.

USE & APPL. The Theory of Proportion in Arithmetic and Algebra is founded on a similar truth; namely,

If four magnitudes be proportionals, and if A, B, C, D, represent those magnitudes numerically, i.e., if A and B represent the numbers of times, the unit of their kind is contained in the two first, and if C and D represent the numbers of times, the unit of their kind is contained in the two last, then the quotient or fraction A shall be equal to ; and conversely," А See Geom. Plane, Sol. and

B

Spher. p. 46 & 47.

70

:

=

C 10: D 6;

componendo, A 5 + B 3 : B 3 = C 10 + D 6 : D6, by 18, V.

If A 5 B 4 C 10: D 8;

convertendo, A5: A 5-B4C 10: C 10-D 8. by Pr. E,V.