MATHEMATICS. MENSURATION. [OONTB-TTS or BARTH oirmxas, KTO. 



AM.P.Di.^Lt. (a + 4p, +,,): 



and the volume of the portion P a M.. M t P 4 is 

 and 8 on ! 



and t)ie volume of Pj.Mj.BCu -.(p,. + 4/>+i + &) 

 , by addition, the whole volume required is 



J. j (a + 6 + 4 (/>,+/>,+...+;>,,.+!) + 2 0>,+J>* 



which may be expressed as a rule as follows. 



ween the first and last sections make an odd num- 

 ber of sections at equal distances along the road, the 

 planes of the section being perpendicular to the road ; 

 thfii one-third part of the common distance multiplied 

 by the sum of the first and hist sections, with four times 

 the sum of the odd sections, and twice the sum of the 

 even sections, gives the volume between the first and last 

 section. 



The student will observe that this rule is the same as 

 that for finding the area of a figure bounded by a curve, 

 which has been already given (p. 647), excepting that the 

 ordinates in the former rule are replaced by trapezoidal 

 sections in the latter. The formula given in article 10 is 

 called the prismoidal formula. It will be observed that 

 the material on each side of a cutting being generally the 

 same, is the reason the inclination of the planes A G 

 and B H to B D being generally the same as stated in 

 the last article. Mr. Macneill, to whom the prismoidal 

 formula is due, has constructed tables founded on that 

 formula, by which the volume of a cutting is very readily 

 calculated. The volume of an embankment is to be 

 found by the same formula, since, as a question of mere 

 figure, an embankment is only an inverted cutting. If 

 the calculation be made directly from the formula, it 

 is very tedious ; the value of the tables above referred 

 to is therefore very great, and is enhanced by the follow- 

 ing circumstance : In constructing a long line of rail- 

 way, the earth taken out of the cutting should be suffi- 

 cient to form the embankment, otherwise land must be 

 purchased for the mere purpose of obtaining earth ; to 

 effect this end of making the volume of the embank- 



! ments equal that of cuttings, the ascents and descents 

 (gradients) of the line of road have to be properly chosen, 



l and this can only be done by trial, so that the calculation 

 may have to be performed two or three times before 

 a right adjustment can be hit upon. It is worth adding 

 that, as a general rule, a cutting is followed not by a 

 long level, but by an embankment ; if possible, these two 

 are adjusted to each other, to prevent the need of carry- 

 ing earth from long distances. 



12. To find the Solid Content of a Military Earth-work. 



The form of a military earth-work will be understood 

 from the following explanations : 



The form of a section of the work made by a vertical 

 plane perpendicular to the face of the work is such as 

 A B C D E F : from B, C, D, E draw perpendiculars to 

 AF, viz., Bm, Cn, Dp, 

 Jv/. then Am, mn, np, 

 pq, q, are of known 

 magnitudes, as also 

 are Bm, Cn, D;>. 

 The plan of the work 

 will be of the accom- A 

 kind, viz.,F'/ 



I'ii 



(Fig. 47) is the lino corresponding to F (Fig. 46), E'e the 

 line corresponding to E, and so on for the others ; the 

 lengths of all these lines are known ; we will call them 

 a, ft, e, d, e,f, respectively. 



In Fig. 46, join pE, pO, pB, dividing the section of the 

 work into triangle* Aj>B, Bj>C, CpD, DpE, and EpF ; 



call these areas respectively A l A g A s A 4 A 6 , then the 

 contents of the work 



are clearly equivalent 

 to the frustums of five 

 prisms (similar to that 

 in the CoroL to article 

 7), which have the area 

 of perpendicular sec- 

 tion A 5 ,andedges/,F,(i, 

 section A , and edges 

 e, d, d, section A.,, and 

 edges *l, d, e, section A 

 and edges c, d, b, and 

 section A, and edges 

 6, </, a, it being evident 

 that the edge through 

 p=d. Hence if the 

 whole volume equal 

 V, we have 



Fig. 47. 



+ 3 * ( + d + 6) + 3 '- (6 + d + a) 



/. 3 V = a A, + 6 (A, + A,) + c (A 3 + A s ) 



+ d(A 3 + A,) + c (A 4 + A.) +/ A. ) 



In which formula it will be observed that each line in 

 the plan is multiplied by the triangle, or by the sum of 

 the triangles, which have an angular point in that line, 

 and that the line dD' is also multiplied by the whole 

 area of the section. Hence if these products are formed 

 and added together, the required volume is one-third of 

 the sum. 



13. To find the Volume of the Frustum of o right Cone 

 made by a Plane parallel tc the Bate. 



Let A B C D be the frustum of the cone PCD. Join 

 P O Oj where O Oj are the centres of the ends of the 

 frustums. Let AO = r, G O x = r x> OO 2 = h. PO = x. 



Then volume ABP 



PCD: 



3 



.'. Volume of frustum 



Now x + h : TJ : : x : 

 .'. h : x : : r x r : r. 



.'. Volume of frustum 



irh , . 



r 



which is the same as the volume of three cones whose 

 common height is h, and which have the radii of their 

 bases respectively r, r, and a mean proportional between 

 r and r, . 



14. To find the Volume of a Portion of a Sphere. 



Let A B O be a quadrant of a circle, draw B C, A C, 

 tangents at A , and B, join O C, and take any point P in 

 A B and draw Q P N parallel to B ; then if the whole 

 figure revolve round A O, the quadrant ABO will 

 clearly describe a hemisphere, the square A O B C 

 will describe a cylinder, on a base whose radius is 

 O B, and height A O, and the triangle AGO will de- 

 scribe a cone on a base whose radius is A C and height 

 A O. Also A P N will describe a portion of a sphere, 

 A C Q N a cylinder whose height is A N and radius A C, 

 and A C M N a frustum of a cone, the radii of whose 

 ends are A C and MN, and height AN. Divide AN 



