S P H 



S P H 



to the diameter A B of the generating circle ; and the dia- 

 meter of a circle, as F E, that does not pafs through the 

 €entre, is equal to fome chord of the generating circle. 



Hence, 2. As the diameter is the greateft of all chords, 

 a circle palling through the centre is the greateft circle of 

 the fphere ; and all the relt are lefs than the fame. 



Hence alfo, 3. All great circles of the fphere are equal 

 to one another. 



Hence alfo, 4. If a great circle of the fphere pafs 

 through any given point of the fphere, as A, it muft alfo 

 pafs through the point diametrically oppofite to it, as B. 



Hence alfo, 5. If two great circles mutually interfeft 

 each other, the line of feftion is the diameter of the fphere ; 

 and therefore two great circles interfeft each other in points 

 diametrically oppofite. 



Hence alfo, 6. A great circle of the fphere divides it 

 into two equal parts or hemifpheres. 



2. All great circles of the fphere cut each other into two 

 equal parts ; and, converfely, all circles that thus cut eacli 



■ other are great circles of the fphere. 



3. An arc of a great circle of the fphere, intercepted 

 between another arc H I L [Jig. 10.), and its poles A and 

 B, is a quadrant. That intercepted between a lefs circle 

 J) E F, and one of its poles A, is greater than a quadrant ; 

 and that between the fame, and the other pole B, lefs than 

 a quadrant ; and converfely. 



4. If a great circle of the fphere pafs through the poles 

 of another, that other pafles through the poles of this. 

 And if a great circle pafs through the poles of another, the 

 two cut each other at right angles ; and converfely. 



5. If a great circle, as A F B D, pafs through the poles 

 A and B of a lefler circle D E F, it cuts it into equal parts, 

 and at right angles. 



6. If two great circles, AEBF and CEDF, (fg. 11.) 

 interfeft each other in the poles E and F, of another 

 great circle A C B D ; that other will pafs through the 

 poles I and i, H and h, of the circles AEBF and 

 CEDF. 



7. If two great circles, AEBF and CEDF, cut each 

 other mutually ; the angle of obliquity, A E C, will be 

 equal to the diftance of the poles H, I. 



8. All circles of the fphere, as G F and L K (fg. 12.) 

 equally diftant from its centre C, are equal ; and the far- 

 ther they are removed from the centre, the lefs they are. 

 Hence, fmce of all parallel chords, only two, G F and 

 L K, are equally diftant from the centre ; of all the circles 

 parallel to the fame great circle, only two are equal. 



9. If the arcs F H and K H, and G I and I L, inter- 

 cepted between a great circle I M H, and the leftcr circles 

 G N F and L O K, be equal, the circles are equal. 



10. If the arcs F H and G I, of the fame great circle 

 A I B H, intercepted between two circles G N F and I M H, 

 be equal, the circles are parallel. 



11. An arc of a parallel circle, I G, (fg- i.) is fimilar 

 to an arc of a great circle, A E ; if each be intercepted be- 

 tween the fame great circles C A F and C E F. 



Hence, the arcs A E and I G have the fame ratio to 

 their peripheries ; and, confequently, contain the fame num- 

 ber of degrees. And hence the arc I G is lefs than the 

 arc A E. 



12. The arc of a great circle is the fliorteft line which 

 can be drawn from one point of the furface of the fphere 

 to another ; and the lines between any two points on the 

 fame furface are the greater, as the circles of which they 

 are arcs are the lefs. 



Hence, the proper meafure, or diftance, of two places 

 *n the furface of the fphere, is an arc of a great circle intw- 



cepted between the fame. Sec more on this fubjeft in 

 Theodofii Elem. Sphxr. apud Dechales Curfus Mathema- 

 ticus, tom. i. p. 145, &c. 



SPHEROID, Sph^roides, 2ipaiposiJfl;, formed from 

 o-ifiaifa, fphxra, and ulv, Jhape, in Geometry, a folid ap- 

 proaching to the figure of a fphere, though not exaftly 

 round, but oblong ; as having one of its diameters bigger 

 than the other ; and generated by the revolution of a femi- 

 ellipfis about its axis. 



When it is generated by the revolution of the femi-cUipfis 

 about its greater or tranfverfe axis, it is called an oblong or 

 prolate fpheroid : and when generated by the revolutiop. 

 of an ellipfis about its lefs or conjugate axis, an eblatc 

 fpheroid. 



The contour of a dome, Daviler obferves, fliould be half 

 a fpheroid. Half a fphere, he fays, is too^low to have a 

 good effeft below. 



For the folid dimenfion of a fpheroid, multiply con- 

 tinually together the fixed axis, the fquare of the revolving 

 axis, and the number -52359877, or -^ of 3.14159, and the 

 laft produft will be the fohdity : i, e. ^pttc = the oblate, 

 and iptcc =z the oblong fpheroid, where /> = 3.14159, 

 / =: the tranfverfe, and c = the conjugate axis of the gene- 

 rating elh'pfis. Or, multiply the area of the generating 

 ellipfe by i' of the revolving axis, and the produft will be 

 the content of the fpheroid : i. e. \tA = the oblate, and 

 I c A. = the oblong fpheroid ; where A is the area of the- 

 ellipfe. E. g. Required the content of an oblate, and of an 

 oblong fpheroid, the axes being 50 and 30. Thus, 50 x 

 30 X .78539816 — 1178.09724 = the area of the ellipfe. 

 And 1 178.09724 X ! X 30 = 23561.9448 = the oblong 

 fpheroid: and 1 178.09724 x 4 X 50 = 39269.908 = the 

 oblate one. 



Dr. Hutton has demonftrated the rule above given in 

 the following manner. Put / = B I the fixed femi-axis, 

 {Plate XIV. Geometry, fg. ■^.) r = I M the revolving 

 femi-axis of the fpheroid, d = S I any femi-diameter of the 

 feftion N B M, b — IK its femi-conjugate, j> = A E an 

 ordinate to the diameter SI, or a femi-axis of the elliptic 

 feftion AFC parallel to K L, and 2 = E F its other 

 femi-axis, alfo x = E I, j = the fine of the angle A E S, 

 or of the angle K I S, to the radius I, and p = 3.14159. 

 Then, by the property of the ellipfe K S L, aa: bb 



, , aa — XN ry 



:: a a — X X : b x = J' J' 5 ^nd * : r : : y : -,- 



aa b 



= 2. But the fluxion of the folid K A C L '\% psyt.x 



psryyx , . . r • , "'y . i 



^- — Y^—t by writing for z its value y, = pbsrx x 



aa — XX , -,„. - , t, aa — XX 

 -, by fubftituting forjjj' its value** x , 



= pfrrx X - — ~ * , by putting for abs its value rf; 



aa — ^ XK 

 and hence the fluent pjrr x x » or t^PJ '''^ >■ 



laa- 



-, will be the value of the fruftum K A C L ; 



which, when E I or x becomes S I or a, gives \pfrr for 

 the value of the fcrni- fpheroid K S L ; or the whole Iplioroid 

 = 'cp F R R, putting F and R for the whole fixed and re- 

 volving axes. Q.E.D. 



Coral. 1 From the foregoing demonftratioD it appear* 



^ T 2 that 



