...it by adding it to the divisor : then multiplying the divisor thus completed by the units, we obtain twice the product of the tens by the units, plus the square of the units. Now if this quantity be greater than the tens of the dividend, the units in the root are too many,...

...is equal to the,tens line, by the unit line BC. But the whole square on AC is made up of the square AE, the two rectangles FE and EC, and the square ED...the square of the tens, plus twice the product of tlie tens Jy the units, plus the square of the units. Let it now be required to extract the square...

...3714 tens, plus two units.) Now the square of the root sought, that is, the proposed number, contains the square of the tens, plus twice the product of...the tens by the units, plus the square of the units. But the square of the tens must give at least hundreds ; hence the last two figures, 44, can form no...

...Subtracting the square of this - - - 2116 Square of 4 tens, or 40 - - - - 1600 516 This remainder contains twice the product of the tens by the units, plus the square of the units. Now, if we double the tens, which gives 80, and divide 516 by 80, the quotient is the figure of the...

...the second period 41, and annexing them on the right of 4, the result is 441, a number which contains twice the product of the tens by the units, plus the square of the units. We may farther prove, as ir the last case, that if we point off the last figure 1, and divide the preceding...

...118'4 0. which sh*>ws thnt the square of a number consisting of (ens and units is composed of tliK square of the. tens, plus twice the product of the tens by the units, j)lus tlic square of the units. This being premised, since the square of a certain number of tens must...

...tens ; hence, we infer that the second power of every number composed of tens and units must contain the square of the tens, plus twice the product of the tens multiplied by the units, plus the square of the units. We will now reverse the process of finding the...

...and (a+4)2=(64)2; or Which proves that the square of a number composed of tens and units, contains the square of the tens plus twice the product of the tens by the units, plus the square of the units. 94. If, now, we make the units 1, 2, 3, 4, &c, tens, or units of the second order, by annexing to each...

...the 7 in the root, and also at the right of the divisor, we multiply by 7, and obtain 469, which is twice the product of the tens by the units plus the square of the units. Hence we deduce the following RULE. *• Separate the number into periods of two figures each, by placing...

...which we bring down the two next figures 84. The result of this operation, 1184, contains twice t/te product of the tens by the units^ plus the square of the units. * But since tens multiplied by units cannot give a product of a less name than tens, it follows that...