___ ___ √_a_ × √_a_ = _a_ ____ √_aa_ = _a_ ___ ___ ___ √_ab_ × √_ab_ = _ab_ ___ ___ ___ ___ ___ ___ ___ ___ (√_a_ × √_b_) × (√_a_ × √_b_) = √_a_ × √_a_ × √_b_ × √_b_ = _ab_ ___ ___ ____ whence √_a_ × √_b_ = √_ab_ 158. It does not follow that a number has a square root because it has a square; thus, though 5 can be multiplied by itself, there is no number which multiplied by itself will produce 5. It is proved in algebra, that no fraction[22] multiplied by itself can produce a whole number, which may be found true in any number of instances; therefore 5 has neither a whole nor a fractional square root; that is, it has no square root at all. Nevertheless, there are methods of finding fractions whose squares shall be as _near_ to 5 as we please, though not exactly equal to it. One of these methods gives ¹⁵¹²⁷/₆₇₆₅, whose square, viz. 15127 15127 228826129 ----- × ----- or ---------, 6765 6765 45765225 differs from 5 by only ⁴/₄₅₇₆₅₂₂₅, which is less than ·0000001: hence we are enabled to use √5 in arithmetical and algebraical reasoning: but when we come to the practice of any problem, we must substitute for √5 one of the fractions whose square is nearly 5, and on the degree of accuracy we want, depends what fraction is to be used. For some purposes, ¹²³/₅₅ may be sufficient, as its square only differs from 5 by ⁴/₃₀₂₅; for others, the fraction first given might be necessary, or one whose square is even nearer to 5. We proceed to shew how to find the square root of a number, when it has one, and from thence how to find fractions whose squares shall be as near as we please to the number, when it has not. We premise, what is sufficiently evident, that of two numbers, the greater has the greater square; and that if one number lie between two others, its square lies between the squares of those others. [22] Meaning, of course, a really fractional number, such as ⅞ or ¹⁵/₁₁, not one which, though fractional in form, is whole in reality, such as ¹⁰/₅ or ²⁷/₃. 159. Let _x_ be a number consisting of any number of parts, for example, four, viz. _a_, _b_, _c_, and _d_; that is, let _x_ = _a_ + _b_ + _c_ + _d_ The square of this number, found as in (68), will be _aa_ + 2_a_(_b_ + _c_ + _d_) + _bb_ + 2_b_(_c_ + _d_) + _cc_ + 2_cd_ + _dd_