1. Reduce 178. 63d. to the decimal of a £. 4 gr. 3. 12d: 6.75 20 s. 17.5625 £ 0.878125 decimal required. 2. Reduce 4 qrs. 1 co. I bu. 3 pec. 0 gal. 2 pts. to the decimal of a last. 3. Reduce 16 cwt. 2 qr. 20 lb. 12 oz. 10 dr. to the EXTRACTION OF THE SQUARE ROOT. The square root of every integral number consisting of either one or two figures, is a single integer: thus the square root of 4 is 2; the square root of 9 is 3; the square root of 64 is 8; and the square root of 99 is 9.94, &c. in infinitum. Again, the square root of an integral number consisting of either three or four figures, will contain two places of integers; and the square root of a number consisting of five or six integers, will contain three integral places. Wherefore, before we can extract the square root of a number, we must ascertain the number of places to be in the root. This is done by separating the given number into periods of two figures each, beginning at the units. place in integers, and proceeding towards the left hand; but at the decimal point in decimals, and proceeding towards the right hand, annexing a cipher, if necessary, to make the number of decimals even. For example, if the square root of 38975 were required, we should form the three following periods: 3|89|75 From which we learn that the square root of 38975 will contain three integral places. In like manner if the square root of 08237956 were required, we should form the four periods following: ·08 | 23 | 79 | 56 And from this we know that the square root of *08237956 must (if finite) be a decimal number of four places. When the given number is partly integral and partly decimal, the process of separating it into periods is the same; for if the square root of the mixed number 137.92145 were required, the periods would appear as below: 137 9214 50 Now these periods indicate that the root will con tain two integral places, and three decimals. When the operation, however, has been carried far enough to obtain these five figures, there will evidently be a remainder, since the last period of the given number is a cipher preceded by a significant figure, a termination which no unit multiplied by itself can produce; and consequently the decimal in the root must run on to infinity: that is, no number whatever multiplied by itself can exactly produce 137-92145. Method of Extracting the Square Root. Having separated the number, whereof the square root is required, into periods, as above directed, we begin at the left hand, and consider what whole number is the greatest square number in the first period. · This number we set under the period, to be subtracted, and write its root for the first figure of the root required. After subtracting the square number from the first period, we annex to the remainder the second period, and for a divisor we double the part found of the root, enquiring how often this double can be found in the tenth part of the augmented remainder. The quotient will in general be the second figure of the root, though sometimes too great. The figure last found is to be annexed to the double divisor, as well as to the part of the root previously found; the whole divisor is now to be multiplied by this figure, and the product subtracted from the proper dividend. Then to the remainder the third period must be annexed, from which the next figure of the root is to be found in the same manner as the second, by doubling the whole root yet found, for a divisor. An example or two will render the method of proceeding more intelligible and easy: let, therefore, the square root of 15376 be required, and the work will appear as under: 153 76 (124 root required. 1 22) 53 44 44) 976- ... Again, let a number be sought which multiplied by itself shall produce 96410-25, and the operation will be, 9164 10 25 (310-5 number required. 9 61) 64 61 6205) 31025 31025 In like manner the square root of 8108.822401 will be found to be 90.049. Lastly, let the square root of the decimal 072 be sought, and the process will be the following: ·07|20|00|00 (2683 &c. in infinitum root required. 4 46)320 528)4400 5363)17600 1511 remainder. TO FIND A MEAN PROPORTIONAL BETWEEN TWO NUMBERS, Multiply the two numbers together, and the square root of the product will be the mean proportional between them. EXAMPLE 1. Required a geometrical mean proportional between 36 and 64. 64 36 384 192 2304 (48 mean required. 16 88)704 704 |