Ill 



AI:K. 



ATIEA. 



t!2 



rounding stars), the nine of which U thus given in the catalogue of 

 Ur published by the Briti*h Association in 1845. 



Annul Dgaiats 

 laK. A. 



rn 



Deel. 

 l'-96 



Formerly the conclusion was sometime* drawn that Arcturus was the 

 nearest star to oar system, from iu being A brilliant Btar with HO 

 decided a proper motion. ThU, which wan but a bint presumption at 

 the time, it now overturn**! by the known fact that there arc much 

 maOer (tan (jt CuawpceiK, fur example) which hare much larger 

 proper motion*. 



ARK, the modern French measure of surface, forming part of the 

 new decimal system adopted in that country after the revolution. It is 

 obtained as follows : the metre, or measure of length, being the forty- 

 millionth part of the whole meridian, as determined by the survey, is 

 3-2809107 English feet ; and the are is a square, the aide of which is 

 10 metres long. The following denominations are also used : 



Decare 

 Hectare 

 Chilare 

 Myriare 

 Declare 

 < ..-. 

 MiUiare 

 The an is 



or 



or , 



i lOares. 

 100 

 1000 

 10,000 

 a. of an are. 



100 square metre*. 

 947-68176 French sq. feet. 

 1076-44144 English sq. feet. 



The hectare is generally used in describing a quantity of land. It is 

 2-4711695 English acres, or 404] hectares make 1000 acres, which 

 disagrees with the first result by less than 1 part out of 50,000. 



A'REA. This term is a Latin word, and means the same thing as 

 Ktperfdtt, or quantity of turf ace, but is applied exclusively to plane 

 figures. Thus we say, " the turf ace of a sphere, the area of a triangle," 

 and " the narface of a cube is six times the area of one of its faces." 

 The word is also applied to signify any large open space, or the ground 

 upon which a building is erected ; whence, in modern built houses, 

 the portion of the site which is not built upon is commonly called 

 the ana. 



Returning to the mathematical meaning of the term, the measuring 

 unit of every area is the square described upon the measuring unit of 

 length : thus, we talk of the square inches, square feet, square yards, 

 or square miles, which an area contains. And two figures which arc 

 timilar, as it U called in geometry, that is, which are perfect copies 

 one of the other on different scales, have their areas proportional to the 

 Kfuara of their linear dimensions. That is, suppose a plan of the front 

 of a house to be drawn so that a length of 600 feet would be repre- 

 sented in the picture by one of 3 feet. Then the area in the real front 

 is to the area of the front in the picture in the proportion of 500 times 

 600 to 3 time* 8, or of 260,000 to 9. Similarly, if the real height were 

 20 time* as great as the height in the picture, or in the proportion of 

 20 to 1, the real area would be to that of the picture as 20 times 20 to 

 once one, or as 400 to 1 ; that is, the first would be 400 times as great 

 as the second. 



Any figure which is entirely bounded by straight lines may be divided 

 into triangles, as in the adjoining diagram. The area of every triangle 



may be measured separately by either of the following rules, in which 

 the word in italics may mean inches, yards, miles, or any other unit, 

 provided only that it stands for the same throughout. 1. Measure a 

 side, A B, of the triangle ABC, and the perpendicular c D which is let 

 fall upon it from the opposite vertex, both in unitt. Half the product 

 of A B and c D is the number of square unite in the triangle ABC. Tlnm. 

 if A B be 30 yards, and c O 16 yards, the triangle contains 240 tjuare 

 yards. 2. Measure the three sides, A c, c B, B A, in twite ,- take the half 

 mini of the three, from it subtract each of the sides, multiply the four 

 result* together, and extract the square root of the product ; this gives 

 the number of square unite in the triangle. For instance, let the three 

 aides be 6, 6, and 7 inches ; the half sum is 9, which, diminished by 

 the three sides respectively, gives 4, 8, and 2. 9, 4, 3, 2, nmJUpHed 

 together, give 216, the square root of which in 14-7, 14& very nearly. 

 The triangle therefore contain* about 14,', mjuare inches, 



The following rules may be applied in the following eases : for n 

 parallelogram, multiply A B, a side, by c D, its perpendicular distance 



from the opposite side; fora rectangle, multiply together adjoining 

 sides, F g and r it ; for a four-aided figure, in which u T and s v are 





parallel, but T v and R s converge, multiply R B, one of the com 

 sides, by T z, its perpendicular distance from the middle point of the 

 other. When R T and a T are perpendicular to R R, then T z is half tho 

 sum of R T and 8 v. 



To find the area of a circle, multiply the radius O A by iUelf and the 

 result by 365; then divide by 113. To find the area of the sector 



A D B, see ANGLE. To find the area of the 

 portion A B D, find those of the sector o A D B, 

 and the triangle O A B separately, and sub- 

 tract the second from the first. In all these 

 cans, the result is in tho ti/uare unit* corre- 

 sponding to the linear unite in which the 

 measurements were made. 



The area of a curvilinear figure can only 

 be strictly found by mathematical processes 

 too difficult to be hero described, but the following method will give 

 an idea of the principles employed. Let A c D B be a curvilinear figure 

 Ixninded by the curve c D and 

 the lines c A, A B, B D, of which 

 the first and third are perpen- 

 dicular to the flecond. Divide 

 A B into any number of equal 

 parts (eight is here supposed) 

 by the pomts 1, 2, 3, Ac., and 

 construct the accompanying 

 obvious figure by making A p, 



1 q, &c., parallelograms. It is 

 plain that the area sought, 

 A C D B, is greater than the sum 

 of the inscribed rectangles, de- 

 noted by the letters or numbers 

 at opposite corners, 



Ic, 2p, 85, 4r, 5>, 6t,7 U,BV; 



and that it is less than the sum 

 of the circumscribing rectangles 



*p, 1 q, 2 r, 8 *, 4 1, S u, 6 r, 7 D. 



Therefore the area sought does not difl'er from either of these sums by 

 so much as they differ from one another ; but the sums differ from one 

 another by the sum of the rectangles 



cp, pq,qr,rt,l, lu,ur,vr>, 



which, placed under one another, give tho rectangle D E, which is less 

 than D 7 : consequently neither -urn ditl'cre from the area sought by so 

 much as D 7. But by carrying the division of A B, with which we set 

 out, to a sufficient degree, the area of D 7 might have been reduced to 

 any extent which might have been thought necessary ; that i. name 

 any fraction of a square inch, however small, and A B can U< divide.) 

 into such a number of equal parts that D 7 shall be smaller than that 

 fraction of a square inch. Hence the sum of the inscribed "r circum- 

 Hcribed parallelograms may, by dividing the line AB sufficiently, be 

 made as nearly equal to the area as any practical purpose can require. 



The accuracy of the preceding process will be increased by summing, 

 not the parallelograms, but the figures 



A c;> 1 , 1 ;,> i? 2, 2 '/ r 3, A.C., 



considering c ;>, pi),<)r, &c., as straight lines. This will be equivalent 

 t<> adding half the rectangle D E to the sum of the rectangles aforesaid. 

 The practical ride in Add all the intermediate ordinates, 1 />. '.!'/. Ac . 

 to the half sum of the extreme ordinates A c and BD : multiply the 

 total by the common value of Al, or 1 2, 4c. This approximation is 

 the first step of the method of QUADRATURES, which see. 



Tho mathematical process of finding the area carrie* the preceding 

 approximation one step further, and finds what is the limit to \\liich 

 the sum of the inscribed parallelograms appro:, ' and nearer, 



as the number of divisions of A B is increw. d. Tliis limit, it is easy to 

 show, is an exact expression for the area required, ll r. pi. . ut one 

 ..f the line* A 1, A 2, &c., and y the correspond] n line 1 //. '2. 7, Ac., the 

 area of the curve is found by the process of the integral calculus thus 

 represented: 



or, in the language of fluxions, 



fluent of >/ .c. 



A process similar to the pmerdiiig is employed by surveyors in 

 measuring a field whose boundaries are curvilinear. [Si in 



r. | 

 The investigation <,f the area of a curve was formerly called tho 



