RULE. Multiply the side of the given square by 99, and divide the product by 70; the quotient will be the diagonal of the square in terms of the side. CASE 2. When the diagonal of any square is known, to find the side. RULE. Multiply the diagonal of the given square by 70, and divide the product by 99; the quotient will be the side of the given square in terms of the diagonal. NOTE. It is believed by the author that the above numbers will answer every purpose for all ordinary measurements; besides this, they are simple and easily remembered by the most ordinary mind. If, however, still greater accuracy is required, attention is invited to the following, which the author ventures to hope will be found to be sufficiently correct to satisfy the most careful. By Case 2, PART 1, it is shown that where the radius of the given circle is the square root of two, the side of the inscribed square will be two, and the side of the inscribed square, the diagonal of which corre99 sponds with the radius, is one. Therefore the secant No. 2 is 70' which, by division, gives the square root of two to five places of figures correct, thus: 1.4142; that is, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into 70 equal parts, the diagonal of the same square will be 99 of the same parts nearly. By Case 3, PART 1, it is shown that the secant No. 3 is 19601 13860? which, by division, gives the square root of two to nine places of figures correct, thus: 1.41421356; that is, if the side of the inscribed square, whose diagonal corresponds with the radius, be divided into 13860 parts, the diagonal of the same square will be 19601 of the same parts nearly. By Case 4, PART 1, it is shown that the secant No. 4 is 768398401 543339720' which, by division, gives the square root of two to 18 places of figures correct, thus: 1.41421356237309504; that is, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into 543339720 parts, the diagonal of the same square will be 768398401 of the same parts nearly. By Case 5, PART 1, it is shown that the secant No. 5 is 1180872205318713601 which, by division, gives the square root of two 835002744095575440 ’ to 36 places of figures correct, thus: 141421356237309504880168872420969807; that is, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into 835002744095575440 parts, the diagonal of the same square will be 1180872205318713601 of the same parts nearly. By Case 6, PART 1, it is shown that the secant No. 6 is 2788918330588564181308597538924774401 1972063063734639263984455073299118880' which, by division, gives the square root of two to 72 places of figures correct, thus: 1414213562373095048801688724209698078569671875376948073176679737990732478; that is, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into 1972063063734639263984455073299118880 sides, the diagonal of the same square will be 2788918330588564181308597538924774401 of the same parts nearly. By Case 7, PART 1, it is shown that the secant No. 7 is 15556130909385807535224779842639686625468648065798177627121099984505505235876371971631359164002982364405164572043531 6514337489817601 7842392159581760 which, by division, gives the square root of two to 144 places of figures correct, thus: 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641563643977195724018929160771077122365330384600627 ; that is, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into 10999845655052358763719716313591 sides, the diagonal of the same square will be 15556130909385807535224779842639686625468648065798177627126514337489817601 of the same parts nearly. By Case 8, the square root of two can be found, by division, to 288 places of figures, by Case 9 to 576 places, by Case 10 to 1152 places, and so on ad infinitum; and, in every case, if the side of the inscribed square, the diagonal of which corresponds with the radius, be divided into the requisite number of parts, the diagonal will still be expressed by a certain number of the same parts till it reaches the vanishing point, when the square root of the sum of the squares of the two sides of any square can be extracted exactly, But if by the word "infinite," indefinite extension only is meant, then the operation may be continued without end, and in THAT CASE there will forever remain an indivisible unit, the square root of which never can be extracted. If, then, the true ratio of the circumference of the circle to its diameter be as 3 is to 1, and the radius of the given circle be the √2, then will the area contained within the given circle be 62, and the side of the inscribed square will be two, while the area will be four. But the area of the circumscribed square is double the area of the inscribed square, therefore the area of the circumscribed square is 8. Again, the square described upon the diameter of any circle is equal to the circumscribed square. But the diameter of the given circle is 18, therefore the area of the circumscribed square is equal to 1/8 X 188. Again, the area of the inscribed square is one-half the area of the circumscribed square; therefore the area of the inscribed square is 4. Now the area of the given circle is 62, and the area of the inscribed square is 4, and the area of the circumscribed square is 8; therefore the area of the circle is to the area of the inscribed square as 11 is to 7; and the area of the circle is to the area of the circumscribed square as 11 is to 14. Therefore to find the area of any circle, when the diameter is known, use the following RULE. Multiply the square of the diameter by 11, and divide the product by 14, the quotient will be the area of the circle; or, square the radius, and multiply it by 34, the product will be the area of the circle. |