So the three continued proportionals are

2:8:32, or

25:15:9.

§ 91. Any equation of this form y 2" + ayTM=b, where the greatest index of the unknown quantity y is double to the index of y in the other term, may be reduced to a quadratic z2 + az = b, by putting yz, and confequently y2 = z2. And this quadratic refolved as above, gives

The product of two quantities is a, and the fum of their fquares b. Qu. the quantities?

Sxy

Supp.

ху

y

[ x2 + y2 = b ...x2 = b —

whence by2=;

mult. by y2.. by2 — y1 — a2,

tranfp. y by2 = — a2.

Put now yz... and confequently y = z1,

and it is

To find a number from the cube of which if you fubtract 19, and multiply the remainder by that cube, the product shall be 216.

Call the number required x; and then, by the question,

361

4

19x3 = 216.

6

Put x3 = z.....x = x2, and it will be

x2-192+ =216 +

361

1225

4

4

But x=Vz; wherefore x + 3, or — 2.

EXAMPLE III.

To find the value of x, fuppofing that x3

But x3 = x2, and x = √≈2 = V/64 = 4.

CHAP. XIV.

OF SURD S.

§ 92. Fa lefs quantity measures a greater fo as to leave no remainder, as 24 meafures 10a, being found in it five times, it is faid to be an aliquot part of it, and the greater is faid to be a multiple of the lefs. The lefs quantity in this cafe is the greatest common meafure of the two quantities; for as it measures the greater, fo it alfo meafures itself, and no quantity can measure it that is greater than itself.

When a third quantity measures any two propofed quantities, as 24 measures 6 a and

10a, it is faid to be a common measure of these quantities; and if no greater quantity measure them both, it is called their greatest common measure.

Thofe quantities are faid to be commenfurable which have any common measure; but if there can be no quantity found that measures them both, they are faid to be incommensurable; and if any one quantity be called rational, all others that have any common measure with it, are alfo called rational: But thofe that have no common measure with it, are called irrational quantities.

§ 93. If any two quantities a and b have any common measure x, this quantity x fhall also measure their fum and difference ab. Let x be found in a as many times as unit is found in m, so that a = mx; and in b, as many times as unit is found in n, fo that bnx; then fhall a bmx ± nx = m ± n x x ; so that a fhall be found in ab, as often as unit is found in m±n: Now fince m and ʼn are integer numbers, m±n must be an integer number or unit, and therefore x must measure a ± b.

;

$94. It is alfo evident, that if x measure any number as a, it must measure any multiple of that number. If it be found in a as many times as unit is found in m, fo that a = mx, then it will be found in any multiple of a, as na, as many times as unit is found in mn; for

na mnx,

$95. If two quantities a and b are proposed, and b measure a by the units that are in m (that is, be found in a as many times as unit is found in m) and there be a remainder c; and if x be supposed to be a common measure of a and b, it fhall be also a measure of c. For by the fuppofition amb + c, fince it contains bas many times as there are units in m, and there is c befides remaining; therefore amb = c. Now is fuppofed to measure a and b, and therefore it measures mb (Art. 94.) and confequently amb (Art. 93.) which is equal

to c.

If c measures b by the units in n, and there be a remainder d, fo that bnc + d, and b-nc-d, then shall x alfo meafure d; because it is fupposed to measure b, and it has been proved that it measures c, and confequently nc, and bnc (by Art. 94.) which is equal to d. Whence, as after fubtracting b as often as poffible from a, the remainder is meafured by x; and after fubtracting e as aften as poffible from b, the remainder dis alfo measured by x; fo, for the fame reason, if you fubtract d as often as poffible from c, the remainder (if there be any) muft ftill be measured by x: and if you proceed, still fubtracting every remainder from the preceding remainder, till you find fome remainder which fubtracted from the preceding leaves no further remainder, but exactly mea

1

fures