Again, because B and B are each prime to An, B2 is prime to A"; and as B and B2 are each prime to A", ... B3 is prime to A"; and in the same manner it may be shown that Bm is prime to A".

that AC E, &c., will be prime to B x D × F, &c.

PROP. VI. THEO.

If one magnitude contain another any number of times, and leave a remainder, such that the greater of the two magnitudes is to the smaller as the smaller is to the remainder, then the two magnitudes will be incommensurable.

Let A, the greater of two magnitudes, contain the smaller, B, any number of times, leaving the magnitude C less than B, such that A : B :: B: C; then A and B are incommensurable, or have not a common measure.

Let the successive remainders, in finding the common measure of A and B, be C, D, E, &c. (prop. ii.)

Because A: B :: B: C, and B does not measure A, ... C cannot measure B; and yet A contains B as often as B contains C. Let s B be the greatest multiple of B which is contained in A; and let tC be an equimultiple of C,

B) A (
s B

which must therefore be the greatest
multiple of C contained in B, or in other terms 8 =

.. As B: B: t C,

C) B (t

t C

D ) C ( u

u D

E, &c.

t;

Now it is evident that D cannot measure, C, because C cannot measure B. Pursuing the same course of reasoning as before, we may readily show that

N

CD: D: E, and that E cannot measure D, because D does not measure C. Consequently the process for finding the common measure of A and B is interminable, .. the magnitudes are incommensurable.*

PROP. VII. THEO.

If a line be divided in extreme and mean proportion, the two parts will be incommensurable.

A line, A, is said to be divided in extreme and mean proportion, when it is divided into two parts, B and C, such that A: B:: B: C.

A

B

C

Allowing this to be the case, A and B are incommensurable, by the last proposition; and therefore B and C are also incommensurable.

PROP. VIII. THEO.

The diagonal and side of a square are incommensurable.

Let A B C D be a square; the diagonal A C is incommensurable with its side A B.

Produce A C, and from the point C, as a centre, with the radius C B, describe the semicircle F BE.

The angle A B C being right, A B touches the circle, therefore (36 B. iii.) A B2; or A E:AB::

A EX AF

=

AB: AF.

And as A E contains A B twice, and

D

C

A

F

E

leaves a remainder, A F; ... A E and A B are incommensurable; hence A C and AB are incommensurable, for if these had a common measure, the same would also measure their sum, A E, which is incommensurable with A B. Therefore the diagonal of a square is incommensurable with its side.

*This proposition, with some alteration, is taken from Professor Young's "Elements of Geometry," Prop. xix, B. v.

Because there are no two numbers that cannot be measured by another (at least by a unit); and as the demonstration of proposition vii. shows that the segments of a line, divided in extreme and mean proportion, are incommensurable, or cannot have a common measure; the lengths of the segments cannot be perfectly expressed by numbers. Again, from the demonstration of the last proposition (viii), it is evident that the side and diagonal of a square cannot be fully expressed by numbers, for they are also incommensurable; and in fact no incommensurable magnitudes can be fully expressed by numbers, or if one of them can be so expressed the others cannot. Hence it might appear, that some very great error should arise if numbers were employed to represent such magnitudes; or that the application of numbers is limited; or that they are not capable of that refined accuracy of expression to which our geometrical notions of things have arrived. The truth is, not one of these conjectures is correct; for we might as well question the accuracy or capability of decimal arithmetic, because the exact decimals of,, §, &c., cannot be wholly expressed. But it is well known that the decimal values of these or any other fractions can be obtained to any degree of accuracy that may be assigned; the same may be said of the application of numbers to incommensurable magnitudes.

To give an idea of the sufficiency of numbers to express incommensurable magnitudes, let us suppose the length of a right line to be 100 millions of miles, and that it is required to express, as near as possible, in numbers, the lengths of the segments of a line of such length, divided in extreme and mean proportion; or to determine, as near as possible, the point which separates the segments. In a very short time we can find that the point which separates the segments is situated in a dot, much less than a period in this print, the distance of which from either of the ends of the line is exactly known-aye, numbers are able to determine that the point which separates the segments is situated in a dot, so small that the greatest microscopic power known could not render it visible; and yet the distance of this dot from either end of the line is exactly known. Next, let us take a square, the side of which let us suppose to be the length of the line taken above, the length of the diagonal of this square can be readily expressed in numbers, so that it will not differ from the true diagonal the breadth of a single hair. The difference between the true diagonal and that which may be expressed by numbers is so small that it cannot be named. And numbers are capable of expressing all other magnitudes, whether they be incommensurable or not, with the same accuracy. Hence numbers cannot be said to be defective either in point of accuracy or power of expression.

SUPPLEMENT II.

ON VARYING AND DEPENDING QUANTITIES.

As the theory of varying quantities is so useful, and so dependant on the doctrine of proportion, at the same time calculated to extend the student's views of that doctrine, a supplement, containing demonstrations of some of the most useful propositions of the theory of variable quantities, must in a great measure contribute to the completion of a work like the present.

In the theory of variable quantities, although only two terms in each proportion are expressed, yet the learner would do well, in such cases, to imagine the other two, for most of our conclusions are come to by considering the properties of proportionals.

DEFINITIONS.

I. A quantity, A, is said to vary directly as another, B; when one of them is changed, the other is changed in the same proportion.

Let A be changed to any other value, a; at the same time B becomes b. Then, if A a :: B : b,

:

A is said to vary directly as B.

The expression A a B signifies, that A varies as B, and is read as such.

Ex.-If the altitude of a triangle do not vary, the area varies as the Lasc. Suppose the altitude of a triangle to be 2 a, and the base B, the area will Le a B; let B change to any other value, b, then the area will be a b. Then it is evident that a B: a b :: Bb; .. the areas vary as their bases.

II. A is said to vary inversely as B, when A cannot be changed in any

, or the reciprocal of B, is changed in the same proportion.

A varies inversely as B, is thus expressed, A α B

This signifies that if A be changed to a, and B to b,

Ex.-If the area of a triangle be given, the base varies inversely as the perpendicular altitude. Let A and a be the altitudes, and 2 B and 2 the bases; .. the areas will be A x B and a × b, as the area is supposed to be

III. One quantity, A, is said to vary as two others, B and C, jointly (thus written, A x B x C); if the former, A, be changed in any manner, the product of the other two, B and C, is changed in the same proportion.

That is, let A be changed to any other value, a, and B x C changed to bc, so that A a B Cbx c; then A is said to vary jointly, as B and C.

Ex. The area of a triangle varies as its base and perpendicular jointly: for let A, B, and P, represent the area, base, and perpendicular of a triangle, respectively; and a, b, and p, those of another; then B× P

=

2 A, and

IV. A is said to vary directly as B, and inversely as C (written thus,

when A cannot be changed in any manner, but B × , or B mul

tiplied by the reciprocal of C, is changed in the same proportion.

That is, let A, B, and C be changed to a, b, and c; and let A: a ::

then A is said to vary directly as B, and inversely as C.

Ex. The base of a triangle varies as the area directly, and as the perpendicular altitude inversely.

Let A, B, and P, be the area, base, and perpendicular of a triangle, and let them be changed to a, b, and p, respectively; then we have, as before, BP = 2A, and b × p 2a;

=