AREAS OP FIGURES.] 



MATHEMATICS. MENSURATION. 



647 



Now, by a property of the parabola 

 Fig. 17. 



Pn P, n, : : pn* :p\ n 2 

 .'. Pn P/, P/ : :p* : p t , * pn* 

 or Pn PH, P/i : : /m 2 : (p l n, pn) (p t n, +pn) 



.'. parallelogram //m t : parallelogram />, : : pit :p l n 



Now in the limit p, n l differs from pn by a quantity 

 less than any that can be assigned. 



.'. In the limit pn : p-f , -\-pn : : 1 : 2 



" In the limit parallelogram pm t : parallelogram 

 pn, : 1 : 2 



which proportion is true for each pair of parallelograms 

 inscribed in P Q R and P Q V, and .' . is true of att, 



.'. In the limit sum of parallelograms in P Q R : sum 

 of parallelograms in P Q V : : 1 : 2 



But the area P Q R is the limit of the sum of the 

 parallelograms inscribed in it ; and P Q V is the limit 

 of the sum of the parallelograms inscribed in it. 



/. PQR :PQV::1 :2 

 .;. PQVR : PQV : :3 : 2 



Now P Q V R is the half of qr QR, and P Q V is the 

 lalfofPQg. 



2- 2 rj. QR. Q.E.D. 



(14). To find approximately the Area of a Plane Figure 

 bounded by a Curve. 



Let A B P Q be the figure ; divide A B into equal 

 parts, AN,, N,N 2) N a 

 N 3 , to., and draw or- 

 dinates P, N,, P, N 2) 

 P, N s , <fcc., parallel to 

 A P or B Q, and perpen- 

 dicular to A B. 



(1). As a first ap- 

 proximation the curved 

 area may be considered 

 as identical with the 

 polygonal area inclosed 

 by P A B Q and the 

 chords PP 1t PjPj, P, 

 P 3 , &o. In this case, T f 



the area required=AN, P t P + N, N. P 2 P, -f N 2 N 3 

 P, P., &c. 



+ N 2 



Or the area= (the distance between any two consecutive 

 ordinates) X (half the sum of the extreme ordinates, 



together with the sum of all the intermediate ordi- 

 nates) 



Orif AN t = A AP=a QB=6 



A BQ p= A 



3 + + y. ) 





(2). The above is a good approximation, but a much 

 better may be found in the follow- 

 ing manner. Consider the portion 

 of the curve between any three 

 consecutive P's such as P 2 P 3 P 4 , 

 through Po draw a line n P 3 m 

 touching the curve, and let it 

 meet N 2 P , N 4 P 4 produced in 

 n and m. Draw P 2 p perpendi- 

 cular to P 4 N 4 , join P 2 P 4 meet- 

 ing P s N 3 in q. Now we may 

 consider P 2 P 3 P 4 a portion of a 

 parabola,* and .'. the curvilinear 



area P 2 P 3 P 4 



Now N 3 g= (y a + V ) and P 2 p 



m 



- of n P a 



.'. P 



N, 



- (y s 



Now the area P a P 4 N 4 N 2 = A (i/ 2 

 /. the whole area P, P 8 P 4 N 4 N s = A (4y 3 +y 2 + y 4 ) 



O 



Hence the area of the figure P A B Q (Fig. 18), which 

 equals P AN 4 P a -f P 2 N P 4 N 4 -(- P 4 N 4 P 8 N e + .... 



f ( 



reckoning that there are 2n + 1 (an odd number) ordi- 

 nates drawn between the two extreme ones PA, Q B. 



.\PABQ=A. 



or the area 



= A x (sum of extreme sides + 4 sum of odd ordinates 



3 

 -(- 2 sum of even ordinates). 



Hence the Rule : At equal distances along the base 

 A B draw an odd number of ordinates parallel to the 

 parallel sides of the figure ; then take the sum of the 

 parallel sides, of four times the odd ordinates, and twice 

 the even ordinates, and multiply the third part of this 

 sum by the common distance between the ordinates, and 

 this product is the area of the figure. 



Besides the above areas of plane surfaces, there are 

 areas of certain solids which will be best given here, viz. , 

 the areas of a prism, a cylinder, a pyramid, a cone, and 

 a sphere. 



DKF. 1. A prism is a solid bounded by two equal 

 and similar rectilinear figures in parallel planes and by 

 parallelograms. 



Thus ABODE, abcde, is a prism bounded by two 

 equal and similar pentagons whose planes are parallel 

 (ABCDE, and abcde), and by parallelograms AaB6, 



Tt appears from Newton, (Lemma XI., Lect. 1,) that in any curve of 

 finite curvature, Ps q i in the limit as the square of }P. Now it is the 

 characteristic property of the parabola that l*sq, is aa the square of 0P 4 ; 

 hence every curve of finite curvature tends to a parabola as its limit, and 

 so if we take a small arc P 2 Ps P* we may consider it to be a parabolic 

 arc, without making any appreciable error. 



