-the hypothesis is, that if mA' be greater than nA, mB is also greater than nB; if equal, equal; and if less, less and it is to be proved that

A': A:: B': B;

or, which is the same thing, that

A' B'

A

B

then the four quantities A', A, B, B,

are equal to

rA+r'A, A, B, B.

Now, let m be such an integer greater than unity; that mr and mr may be each greater than 2; and take n the next integer greater than mr, of course n will be less than mr+mr';

and the four multiples mA', nA, mB', nB,

become

mrA+mr'A, nA, mrB', nB.

But by construction

mrtmrzn mrA+mr'AnA:

By construction

and therefore

therefore by hypothesis, also mB'\nB ;

so that mB' is both greater and less than nB, which is impossible.

A'

A

It is manifest therefore that cannot be greater

than; and in like manner it is shown that

B'

B

not be greater than and therefore

that is, A': A:: B': B.

SCHOLIUM. Thus we have shown, that if four quantities be proportionals by the common algebraic definition, they will also be proportionals according to Euclid's definition; and conversely, that if four quantities be proportionals by Euclid's definition, they will also be proportionals by the common algebraic definition; and by a similar method of reasoning we may easily show, that when four quantities are not proportionals by one of these two definitions, they cannot be proportionals by the other definition.

Thus it appears, that the two definitions are altogether equivalent; each comprehending, or excluding, whatever is comprehended, or excluded, by the other.

THE END.

ERRATA.

Page 6, line 21, for letter, read letters.

I

p. 19, l. 1, 3, for 8xy and 55, read 3xy and 55x.
p. 33, l. 12, for to, read to the addition of.

p. 44, l. 20, for 5a, read 5a3.

p. 83, l. 22, 23, for (m −n) and (m±n), read (m—
n)c and (min)c.

do. 1. 26, for members, read numbers.

p. 88, l. 34, for continued, read contained.
p. 96, l. 15, for ac2, read a3 c2.

p. 104, l. 11, 12, for +63 and -b3, read +62 and

p. 116, l. 12, for 3x, read 3ax.

p. 125, l. 3, dele 2,.

p. 132, l. 2, 9, for 4x2 and +2ub, read Cx and
+2bx.

p. 135, l. 7, for a3, read a2.

p. 137, l. 13, for =

a2

read
"
1 α

a2

1 -α

p. 138, 1. 22, for law that, read law, that is to say,
at the general formula, that.

p. 156, 1. 7, for plynomials, read polynomials.
p. 159, l. 16, for (a+x2), read (a+x)3.

p. 161, l. 12, for unknown, read known.

p. 168, l. 7, for conditions, read condition.
p. 173, l. 4, for x=2, read x= 4.

p. 176, l. 14, for 30, read 20.

p. 179, l. 7, for 198, read 199.

p. 180, 1. 8, for quantities, read quantities, values.
do. 1. 29, for formulæ, read formula.

p. 181, l. 5, 16, for formulæ, read formula.

du. 1. 21, dele 199.

p. 188, I. 17, for 9, read 13.

p. 192, 1, 5, 11, for 2x and +-61, read 3x and G..

p. 195, l. 21, for quantities, read equations.
p. 200, l. 15, for are, read are the.

p. 201, l. 13, for furnish, read furnishes.

p. 215, 1. 5, for Analysts employ, read Analysis
employs.

p. 216, 1. 2, for formule, read formula.

p. 224, 1. 2, for depend, read depends.

p. 248, l. 15, 18, for y= and y=3, read y=3 and
y=2.

p. 254, l. 19, for 8, read x.

read_8y+5

18

p. 256, 1. 2, 21, for 9 and +100, read 4 and y4

100.

p. 257, 1. 23, for other, read others.

p. 279, 1. 7, for form, read from.

p. 284, l. 12, 27, for plus and 720, read minus and

752.

p. 289, l. 13, for 4, read 5.

p. 301, l. 11, for 3s. and 2s. read 2s. and 3s.
p. 313, I. 23, for devoted, read denoted.

p. 314, l. 5, 9, for an-3b6 and is, read an-363 and
dele is.

-

p. 317, 1. 14, for 3/2 ̄x3a=x3, &c., read
/27x3a= (Art. 293) 3/27×3 ×3/a° = 3 ×
X x Xa3
=- -3a2 x.

-

p. 323, l. 8, for 16x2y1 and 6y5, read 15x2y1 and
Gxys.

p. 325, l. 25, 26, for square of b2 and root c2, read
square root of ba and root of c2.

do. 1. 31, for +20x3-15x2+5x-1, read -
20x3+15x3-5x+1.

3

do. l. 33, 34, for a 4a3x, read a-4a3x,
and for 4ax+3, read 4ax2+.

p. 329, l. 3, for square of, read square root of.

a+a3,

a+

p. 349, l. 5, 6, for a* and a3 +a3, read a3 and a

3

do. 1. 18, 30, for 6/243 and 10/9, read 243
and 23/9.