aa + ab + bb a3 a b+ab2 -a2b—ab”—b3 From these operations it appears, 1st. That the square of the sum of any two quan: tities is equal to the sum of the squares of those quantities increased by twice their product; which answers to the 4th proposition of the 2d book of the Elements. 2dly. That the cube of their sum is equal to the sum of their cubes increased by the quantity that arises by multiplying their sum into three times their product. For 3a2b+3ab2 is=a+b×3ab. : 3dly. That the square of their difference is equal to the sum of their squares diminished by twice their product. This conclusion corresponds to the property of geometrical squares and rectangles, demonstrated in the 7th proposition of the 2d book of EUCLID for it is there proved, that if any line denoted by the letter a be divided into two parts 6 and ab, the square of the whole line a, together with the square of the part b, is equal to twice the rectangle contained under the whole line a and the part b, together with the square of the other part ab; that is, aa+bb=2ab+a-6; consequently a-bisaa —2ab+bb. Therefore, 4thly. The sum of the squares of any two quantities is greater than twice their product. 5thly. The cube of the difference of any two quantities is equal to the difference of their cubes diminished by the quantity which arises by multiplying their difference into three times their product. For a—b3 (=a3—3a2b+3ab2—b3—a3—b3—3a2b—3ab2) is=a3-63-3abxa-6. Therefore, 6thly. The difference of their cubes must always be greater than that quantity, or a3 3 must always be greater than 3abxa-b. 7thly. The product of the sum, and difference of any two quantities, is equal to the difference of their squares. This answers to the 5th proposition of the 2d book of EUCLID; for it is there proved, that if any line, as AB, is divided into two unequal parts AD, DB, and also into two equal parts AC, CB, FIGURE IV. the rectangle under the two unequal parts AD, DB, together with the square of the line CD that lies between the sections, is equal to the square of the half, or of CB. Now, 'tis evident that CD is less than CB, that AD (the greater of the unequal parts) is equal to the sum of AC and CD, or CB and CD, and DB (the lesser of the unequal parts) to their difference; therefore CB+CD×CB−CD + CD2 is = CB2; and consequently CB+CD × CB—CD=CB* -CD2. It may be useful to observe, that this proposition of EUCLID proves that if a given line AB be supposed to be described by the motion of the point D from B towards A, the rectangle under the segments AD, DB, will increase continually, till the point D coincides with the point C; for so long the square of CD decreases: afterwards this rectangle will decrease continually while D is passing from C to A for, during that time, the square of CD increases again till it equals the square of CA: and the greatest magnitude this rectangle ever attains is the square of CB, or of half the given line AB. 8thly. The product that arises by multiplying together the sum of two quantities and the sum of their squares diminished by their product, is equal to the sum of their cubes. And, 9thly. The product that arises by multiplying together the difference of two quantities, and the sum of their squares increased by their product, is equal to the difference of their cubes. These, and other like observations, are very frequently of use in algebraic computations. 19. In division the rule for determining the signs of the terms of the quotient is as follows: "Whenever the dividend is marked with the sign +, the quotient must be marked with the same sign as the divisor; and whenever the dividend is marked with the sign the quotient must be marked with the sign that is contrary to that of the divisor." For the quotient multiplied into the divisor must produce the dividend. 1 ARTICLE XXVI. A View of the Diophantine Algebra. By Robert Adrain. 1. We may define the general object of this inquiry to be the method of finding such rational values of one or more indeterminate quantities, that any proposed functions of those quantities may become rational squares, or cubes, &c. It will save frequent repetition to remark here, that all the numbers given or required in this subject are understood to be rational. 2. Let us begin with the formula ax+b; a, and b, being any given numbers. As an example of this formula, suppose it be required to assign such a value for x that 2x+3 may be a square number. Here, if we substitute 1 for x, 2x+3 becomes 2+3 or 5, which is not a square, and therefore I is none of the required values of x; if x=2, we have 2x+3=7, which is not a square; but if x=3, we shall have 2x+3=6+3=-9 which is evidently a square number, and therefore 3 is one of the required values of x. It is easy to find as many of these values of x as we please, merely by equating the proposed formula successively to any number of known squares. Thus, suppose 2x+3=25, whence 2x=22, and therefore x=11, which is one of the values sought. But we may have a general expression for all the values of x by assuming 2x+3=a2; from this we have 2x=a2-3, and x= a2-3 2 •: we may now assume a of any value at pleasure, and we shall obtain successively all the values of x that can render 2x+3 a rational square. On the very same principle we may find in a general manner all the possible values of x that will render ax+b a rational square; we have only to suppose c2-b ax+b=c2, from which we obtain x= T3. Let. ax+b cx+d a · be made a square, a, b, c, and d, being any given numbers. This formula is equally easy with the preceding; we have only to suppose r=the root of the square, which furnishes us with the equation ax+b from cx+d which is a general expres sion for all the possible values of x that can answer the question, by assuming for r successively all pos¬ sible numbers. ax-tb cx-d is made Any particular case of the formula a square in the same manner; for example, let us 1+x find the values of x which will make a square. Assume 1+x 1- X =r2, and from this we obtain x= in which r may be taken at pleasure: if we suppose T 1 which is a square, 4. Let (ax+b)×(cx+d) be made a square; a, b, c, and d, being as usual given numbers. This formula is reducible to that in the preceding paragraph by remarking that any square multiplied or divided by a square will always produce a square: if therefore (ax+6)×(cx+d) be divided by (cx+d2), the will always be a square when the proposed formula is a square, and vice versa. There quotient ax+b cx+d ax-b fore we have only to assume r2, and con cx+d bdr2 sequently x=- as in the preceding paragraph. But the formula (ax+b)×(cx+d) may be made a square without reducing it to ax+b cx+d this is effected by supposing the root of the square to be (cx+d)xr, which gives the equation (ax+b)y (cx+d)=(cx+d)2 Xr2; dividing by cx+d we have ax+b=(cx+d)×r2, b-drz and consequently x= 2 +1 as before. As an example of this, let 1-x2 or(1+x)×(1−x) be made a square. Assume (1+x)` (1−x)=(l—x)3 Xr2, dividing by 1-x we have 1+x=r2 r2x, and therefore x= and r may be taken at pleasure. Suppose r=2, this gives x=2, and 1-3=10=a square; or suppose r=3, which gives x=4. In the same manner we may assign the values of x which will make x2-a2 a rational square, a being any given number. 5. Let the formula of the second degree ax2 + bx + c be made a rational square. We commence this solution with remarking that |