Now, by the table 38=6,614414, and this multi plied by 45-277,39863 the root required What is the square root of 9675? Answer, 98 3615775. Required the square root of 19350. Answer, 139 1042775. When the square root of any number cannot be so reduced as to obtain the answer from the table, as above directed, place the number down, the square root of which is to be extracted, and point it off by pairs as in the common method, then find the square root to the second or third point from the left hand by the tables; double the root thus found for a divisor, to which annex the next figure in the quotient, in the usual way, and then proceed on with the operation as in the common method. EXAMPLE. What is the square root of 768437? Here the nearest square to 76 the first point lies between 8 and 9; look from 80 downwards and you will find 87 to be the nearest to 7684. An easy method to prove the square root.-Cast the nines out of the remainder when the work is finished; subtract NOTE. I have inserted the above as a specimen of what may be obtained by the squares in the table; for sometimes three figures in the root may be obtained at once; nor does it the excess from the number to be extracted, rejecting the nines. Again reject the nines from the root, and square that excess; if both are the same it is right; thus, in the remainder above, the excess more than the nines is 8; I therefore subtract this 8 from the number extracted; and by rejecting the nines there remains 0. In the root, after rejecting the nines there remains 0, which being squared produces 0; and therefore the operation is right. What is the square root of 411? After pointing off from the units' place as before directed, the dot will be over the 4; and, as there are two dots, it shows that there will be two whole numbers in the root; the remaining figures in the root will be decimals. The square root of 4 being two; look to the table from 200 downwards till you arrive as near as possible to 411, and you will find that opposite to 202 stands 40804; but as 202 consists of three figures, the last must be made a decimal; and then proceed on with the remainder of the work in the following manner. 411 (20.2731 40804 4047) 296 40543) 1271 X3 121629 405461) 5471 X1=405461 Here the pairs of ciphers on the right hand of the work may be omitted; it will save both time and trouble. matter whether the number to be extracted be mixed or not. The whole numbers may be ascertained by the dots over the number to be extracted; for example, What is the square root of 176,60? In the above there are two dots over the whole numbers; and therefore there must be two whole numbers in the root; and, if the operation be carried on, all the rest will be decimals. The nearest square root to 176 is 13; but as we would have 3 figures in the root; turn to 130 in the table, and proceed downward till you find the nearest square to 17660, and it will be 132: strike off the last right hand figure 2 for a decimal, and proceed, as above, with the operation. What is the square root of 565 ? Answer, 23.7697+ Answer, 26-2106+ When the square root of decimals is required, care must be taken that they are made even previous to pointing them off: thus, for example, I want to know the square root of the decimal 7; by adding a cipher I have 70, which is even; the square root of which (see the tables) is 8,3666003; but, as the number to be extracted is a decimal, consequently, the whole root is decimals, namely, instead of 8,3666003 write ,83666003, which is the root required. Again: What is the square root of 2, or 2,5, or, which is the same thing, 2,50? Here look to the table for 250, the root of which is 15,8113885; but as there can be only one whole number in the root, the answer will be 1,58113885, the square root required. From these considerations I am induced to set the following examples, the answers to which may immediately be obtained by consulting the table as before directed. What is the square root of 95 ? Answer, 97467943. It is required to extract the square root of 2,1. Answer, 1,44913767 It is required to extract the square root of 14700, Here, by the table, the square root of 147 is 12,1243557; and, because there is an addition of two ciphers to 147, it requires that there shall be one whole number more in the root; and, therefore, by taking in the first left hand decimal for that whole number, the rest will remain for decimals; therefore, 121,243557 is the square root required. Required the square root of 1360000. Here, by the table, the square root of 136 is 11,6619038; but, as there are four ciphers annexed to 136, it requires that there shall be two more whole numbers in the root than the above, and the rest will remain as decimals; therefore, 1166,19038 is the square root required. EXAMPLES. What is the square root of 17300? Answer, 131,592464. Required the square root of 1850000? Answer, 1360,14705. When the square root of any number is given, to find the square root of any other number of times of that number. Rule. Multiply the square root of that number given by the square root of the number of times given, and the product will be the square root required. EXAMPLES. The square root of 7 is 2,6457513; what is the square root of 3 times 7, or 21? Here, by the table, the square root of 3 is 1,732058. Therefore, 1,732058 × 2,6457513=4,58257565576604, the square root required. The square root of 126 is 11,2249721, what is the square root of 504, or 4 times 126? Answer, 22,4499442. The square root of 151 is 12,288057, what is the square root of 1359, or 9 times 151? Answer, 36,864171. The square of any number being given, to find the square of the next number. Rule. To the square of the number given add twice that number plus 1, and it will give the square of the next number required, thus: The square of 35 is 1225 (35 × 2)+1 = 71 1296 36 × 36, or 36o Now, the square of 36 being found, the square of 37, 38, &c., may be found in rapid succession, by only adding 2 every time to each succeeding number as you proceed on. Thus, the square of 36 being 1296 73=71+2, or 36+37, the root in question, and the preceding. 3721369 75=73+2 382-1444 77=75+2 392=1521 79=77+2 402-1600 From the above it may be seen that almost ten square numbers following in succession may be obtained in the short space of one minute. |