the whole figure, we are to add them together, and it will be, 4048.69 area of AEB. 1386.00 area of EFB. 1685-21 area of BFGH. Sum 9849-02 WHOLE AREA. Wherefore, W.G. divisor 231) 9849-02 (42-63 wine gallons. A. G. divisor 282) 9849-02 (34.92 ale gallons. M. B. divisor 2150) 9849-02 ( 4.58 malt bushels. To determine the Content of a Circular Area an Inch in Depth. Before we give directions for solving this problem, we think it right to shew the manner of obtaining the Circular Divisors, and the Circular Gage-points. And to enable the learner to understand this subject clearly, it will be necessary to enlarge a little more on the circle. It has already been stated, in the First Section. of Mensuration, that no rectilineal figure whatever can be constructed precisely equal to the circle, because, although a parallelogram in length equal to half the circumference, and in breadth equal to the radius of the circle, is in area equal to the circle, yet as no finite number will express the breadth of the parallelogram when the length is finite, nor the length when the breadth is finite, the parallelogram, reduced to numbers, can exist only in idea. Notwithstanding the impossibility, however, of squaring the circle, the side of a square that shall differ in area extremely little from the circle, is easily found, and for every practical purpose this is sufficient. It is evident that the square root of the number of square inches in any area, regular or irregular, will be the side of a square of equal area; since the side multiplied by itself must produce the area: hence, if the area of a circle be 1, the side of the equivalent square will be 1. Or if the diameter of the circle be 1, the area will, as before mentioned, be 785398, and the side of the equal square will be the square root of that decimal, or 886226. This being premised, we shall point out the relation between the diameter of a circle and the side of its Inscribed Square, which thoroughly to comprehend is essential in the theory as well as practice of Gaging. CIRCLE AND ITS INSCRIBED SQUARE. B Diameter 1. C D Let ABCD be a circle, of which BC is the diameter; and let ACDB be a square inscribed in the circle, having for its diagonal BC. Then it is plain that the diameter of the circle is equal to the diagonal of the inscribed square. Again the triangles BAC, and BDC are both rightangled. But Euclid, in the 47th proposition of his First Book, has demonstrated that the square described upon the side opposite to the right angle, is, in every right-angled triangle, equal to the sum of the squares described upon the sides containing the right angle. Now, in either of the triangles BAC and BDC, the sides about the right angle are equal; hence the square of the diagonal is double the square of the side. That is, if the diameter of the circle be 1, the side of the inscribed square will be the square root of the half of 1, or 70716 &c. In other words the square root of twice the square of the radius of the circle is invariably equal to the side of the inscribed square. As it is exceedingly convenient to know all the proportions of the circumference, equal square, inscribed trigon, inscribed square, and area, when the diameter is unity (since circles (and consequently similar figures described in them) are always to one another in the direct ratio of the squares of their diameters) the numbers representing them are here stated in order. If the diameter of a circle be 1, The circumference 3.1416 Side of the square equal 8862 Then Side of the inscribed trigon is 8660 .7071 Area .7854 The numbers thus carried to four decimal places only, are sufficiently correct for ordinary purposes, and should be carefully committed to memory. That the young practitioner may exercise himself in these proportions, and thus become better acquainted with them, we subjoin, solely for this end, the lengths of the several lines, and the area, when each is unity in succession. 1. When the Circumference is 1. The diameter is....... -3183 2756 The side of the square equal................ •2821 The area ..... ⚫0796 2. When the Side of the equal Square is 1. 3. When the Side of the inscribed Trigon is 1. The circumference is The diameter 3.6276 1.1547 The side of the square equal.......... 1·0232 Side of the inscribed square........................ Area...... .8165 1.0481 4. When the Area of the inscribed Square is 1. Side of the square equal........... Side of the inscribed trigon............................ Side of the inscribed square...................... 100 ⚫10000 •9772 -7979 |