GEOMETRY. 



measured by the area of its 

 base multiplied by J of its 

 height SO. If now we double 

 indefinitely the number of 

 the faces of the pyramid, the 

 volume of the pyramid ap- 

 proaches nearer and nearer to 

 the volume of the cone, and 

 ultimately coincides with it ; 

 the base of the polygon ap- A 

 preaches nearer and nearer 

 to the base of the cone, and 

 ultimately coincides with it. The height of the 

 pyramid is always the height of the cone ; there- 

 fore the volume of a right cone on circular base is 

 measured by the area of the base multiplied by a 

 third of the height. 



The Sphere. 



It may be shewn that when a triangle revolves 

 round an axis situated in its plane, and passing 

 through the vertex without 

 crossing its surface, the 

 volume generated is equal 

 to the surface generated 

 by the base multiplied by 

 one-third of the altitude. 

 Hence it follows that the 

 volume generated by a 

 sector AOB of a circle 

 turning round one of its 

 sides OB is measured by 

 the area which is generated by the arc AB, mul- 

 tiplied by one-third of the radius of the circle. 

 Therefore, when the arc 

 AB becomes a semicircle, 

 the volume of the sphere 

 is equal to its surface mul- 

 tiplied by one-third the 

 radius. Let V denote the 

 volume, S = surface of 

 sphere, we have 



v = s- r r 



Hence to find the volume of a sphere, we multiply 

 four times the cube of the radius by 3}, and divide 

 the product by 3. 



CONSTRUCTION OF SCALES. 



In practical geometry, scales of various kinds 

 are used for the construction of figures. Scales 

 are lines with divisions of various kinds marked 

 upon them, according as they are to be used for 

 measuring lines or angles. The name of scales is 

 given to lines so divided, because the Latin word 

 for ladder is scala, and the divisions are equi- 

 distant like the steps of a ladder. A line so 

 divided is for the same reason said to be gradu- 

 ated, this word being derived from the Latin 

 jrradus, a step. 



The values of the magnitudes of lines or angles 

 are numbers representing the number of times 

 that some unit of the same kind is contained in 

 them. 



The unit of measure for lines is some line of 

 given length, as a foot, a yard, a mile, and so on. 



The unit of measure for angles is the ninetieth 

 part of a right angle. 



The method of constructing a scale of equal 

 parts is the following : 



Lay off a number of equal divisions, AB, BC 

 CD, &c., and AE, and divide AE into ten equal 

 parts. When a large division, as AB, represents 



10, each of the small divisions in AE will repre- 

 sent i. When each of the large divisions repre- 

 sents 100, each of the small divisions in AE 

 represents 10. Hence, on the latter supposition, 

 the distance from C to n is 230 ; and on the 

 former supposition it is 23. 



If the large divisions represent units, the small 

 ones on AE represent tenths that is, each of 

 them is ^V> r ! On this supposition, the distance 

 C is 2-3. 



To construct a plane diagonal scale. 

 i. A diagonal scale for two figures. 

 Draw five lines parallel to DE, and equidistant, 

 and lay off the equal divisions AE, AB, BC, CD, 



&c., and make EP, AQ, Br, C2, &c., perpendicu- 

 lar to DE. Find m, the middle of AE, and draw 

 the lines Qtn, mP. 



The mode of using this scale is evident from the 

 last. If the large divisions denote tens, then from 

 to o is evidently 34. 



2. A diagonal scale for three figures. 



Draw ten lines parallel to DE, and equidistant 

 Lay off the equal parts, AB, BC, CD, &c., and 

 AE, and draw EP, AQ, Bi, C2,... &c. perpen- 

 dicular to DE. Divide QP, AE into 10 equal 

 parts. Join the ist, 2d, 3d,... divisions on QP 

 with the 2d, 3d, 4th,... divisions on AE re- 

 spectively. 



t 



If the divisions on AD each represent too, each 

 of those on QP will represent 10. Thus from 3 on 

 AD to 8 on QP is 380 ; but by moving the points 

 of the compasses down to the fourth line, and 

 extending them from n to o, the number will be 

 384. For the distance of 8 on QP from Q is 80, 

 and of r from A is 90 ; and hence that of o from 

 the line AQ is 84. 



When the divisions on AD denote tens, those 

 on QP denote units ; and from n to o would then 

 represent 38^, or 38-4. 



When the numbers representing the lengths of 

 the sides of any figure would give lines of an 

 inconvenient size taken from the scale, the num- 

 bers may be all multiplied or all divided by such 

 a number as will adapt the lengths of the lines to 

 the required dimensions of the figure. 



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