i IKi'I.E, ASTRONOMICAL. 



CHICLE, ASTRONOMICAL. 



aid 8 ii, if the lino B 11 be any line drawn through 11. cutting the 

 circle. 



6. If y move nnind Uio circle, and A and P remain fixed, Ui.- angle 

 (opening) A <J P preserves the name magnitude throughout ., namely, half 

 of the angle A o p, and equal to the angle BAP. 



These properties, with several others which are ruilJy true, are in 

 the third book uf Euclid. We name one or two others, the verification 

 of which will be a test of correctness in drawing for those who know 

 how to use the compass and ruler. 



7. From any point T exterior to a circle, two tangent* T \ and T v (of 

 equal lengths) can be drawn. Let T be called the pole of x v. Then it' 

 any number.of poles be taken on the same straight line (which call the 

 polar line), all their chords pass through the same point ; which List 

 point is interior to the circle if the polar line be altogether exterior ; 

 and exterior, if the polar line cut the circle. 



any hexagon, having no opposite sides parallel, be drawn in a 

 circle, the three points of intersection made by lengthening the oppo- 

 site sides must be in one straight line. 



To find the circumference of a circle (with more than sufficient 

 nearness for practical purposes), take the 113th port of 355 time* 

 the diameter (A L) or 3-14159 times the diameter. To find the area in 

 7orr units, multiply the number of units in o A by itself, and take 

 the 113th part of 355 times the result (or multiply by 3-14159). 

 Given the arc A r, and the (radius o A, to determine the angle A o p. 

 [ANGLE.] And the same for the inverse question. To find the area 

 of the sector A o p in square units, take half the product of th.- unit - 

 in the radius and the arc A p. These are the principal questions which 

 can be solved by a person unacquainted with trigonometry. 



The influence of the properties of the circle upon abstract mathe- 

 matical analysis has been so great, that an attempt to describe the 

 manner in which the menus of expression derived from this figure h.ive 

 been used would fill a volume. We can only here give such a descrip- 

 tion as will help the beginner in trigonometry to extend his notions of 

 the symbols he uses. Originally the sine, cosine, 4c. [Ti; 

 MKTRY], meant certain lines drawn in a circle, with reference to a 

 given angle at the centre. Each angle therefore had a sine, &c. for 

 every different length which the radius might be conceived to have. 

 But this introducing an unnecessary complexity into formula;, it wax 

 thought sufficient always to suppose the radius a unit, which wan 

 however always expressed. Thus in the first stage of the science we 

 have this theorem : " The sine of 30 is half the radius," which in 

 course of time took this form, "The radius being 1, sin. 80 is J." 

 This method amounted to defining the sine, &c. to be, not the lines 

 which they originally stood for, but the numerical ratios of these lines 

 to the radius. Thus the sine, cosine, &c. became abstract numbers. 

 The next step was to make the angle itself an abstract number, in the 

 manner which [AXOLB] we have called the theoretical or arcual method 

 of measurement ; that is, instead of measuring the angle by an arbi- 

 trarily chosen angle, such as a degree or a minute, the numerical ratio 

 of the arc to the radius became the measure of the angle. One exten- 

 sion more completed the subject. Angles of more than four right 

 angles were admitted, conceived to be made by the revoluti 

 point, which was considered as having made more revolutions th.ni 

 one. Thus any number represented some angle, and had its sine, 

 oosiue, Ac. And the angles themselves being abstract numbers, and 

 also their sines, &c., it followed that all the propositions of what was 

 trigonometry, an application of geometry, became the propositions of 



trigonometry, a part of pure arithmetic; retaining indeed tl M 



names derived from geometry (names are never changed, witness the 

 use of the term H/uan in algebra), but based upon the notion of 

 number, and thesymbolic operations of algebra. Thus though it will 

 always perhaps be thought desirable to lead the beginner through the 

 gate of geometry, yet there must come a time, if he continue his 

 studies to the higher branches, when he will consider a sine as a 

 number, a function of a number ; for instance, x being t number, the 

 sine of the number i means the series 



_^ 



* 2.3 



2.8.4.5 



- 4c. ad inf., 



or any algebraical form which if equivalent to it. 



This is the point to which works on trigonometry are rapidly 

 tending; and seeing that the student must end, if he pursue his 

 course,_ in such considerations, it is most desirable that he should 1 >cgin 

 in the 'name, manner, to every extent which ii consistent with not 

 forcing abstractions upon him too rapidly. 



The ratio of the diameter to the circumference of a circle i 

 nmr indeed to that of 113 to 355, which numbers this diagram will 



-. - : '.'..'..: \ 



113 I 85 



This method makes the circumfurciiix too great by about twunly- 

 seren hundred-iiiillionths of the diameter, and la therefore aWmdiatly 

 exact. But for common purposes, take the circumference at three 

 times the diameter and one-seventh of the diamct.-r. which does not 

 re than about one part out of a thousand of the diameter, and 

 till in exooM ; that is, the circumference is a little lew than twenty 

 two-sevenths of the diameter. 



CIRCLE, ASTRONOMICAL. Though almost all the astronomical 



and geodeoical instruments which are at present used in measuring 

 angles, are composed of entire circles, the term attrunuuiiral r 

 ordinarily confined to those instruments of which the ~>1, 

 use is the measurement of angles of altitude or zenith distance. In the 

 ("resent article we shall adopt this limitation, and restrict ourselves 

 still further to a description of the construction and use of those 

 which arc cither fixed in the plane of the meridian, as the mural and 

 Imruit circles, or which continue to have the plane of the pi 

 circle vertical, though turning upon an axis, as the aUiluJt and 

 circle. For other instruments which might be included mi- 

 term circular, the reader must consult the articles EQUATORIAL, 

 IIKI-KATLV; C'incLE, and TUEODOLET. 



This article will bit more intelligible with some i>iclituuiary 

 explanations. 



Let a circular plate, divided into 360, turn round a concent i 

 c, fixed into the block 8 K, so that the line u o, moving with tin 

 and in which direction the observer is supposed to look, con be placed 

 in any direction up or down : A and B are two pointers attached t.i th. 

 block, and in a line passing through the centre. The block and circle 

 are supposed to be upright; the axis c and the line A n hori/ontal. 

 Also, when the line of sight is vertical, or E o coincides with z'z, of 

 the divisions ought to be exactly under the [winter A. It is evident, 

 when the line of sight is moved through any angle into the dii 

 EO, that c must have moved through an equal angle, and tl 

 / oo = / A c 0, or the arc A 0. Hence, if an object is seen in the 

 ti K o, its :eith Pittance 7. c o will be the angle pointed out 

 \i]Kin the divided circle by the pointer A. This angle is tech < 

 called the reailimj of A. The line E o represents the direction of plain 

 sights, or the line of sight of a telescope, sometimes cill.d ih< 

 rullimittiun (from cvUimare, said by Facciolati to be a mistni. 

 collintare). 



If, when EO is vertical, A points a little alxiv,- ..r lielow the 0, or 

 zero of the divisions, the difference from is called the . 

 motion, which may be corrected by shifting the position of A; but when 

 the quantity and direction of the error are known, th. ic i- n.. i... .! t 

 make any alteration. For instance, let A point to '2 

 sight is vertical, then it is evident that when the line of sight is in the 

 direction EO, the zenith di 1,111. . w ill IK. e.|iial t<> the arc from - 

 or 2 must be tubtracted from the nadiuy of A. If EO l*'d:. 

 anywhere between /. and .v, then 2 must be a</-t,,l to tin- t. i. lings of 

 A. In fact, the pointer may IK- placed any when- in the circunit 

 of the circle, and the di\ i-i.'iis may commence in any part of the circle 

 without at all tuTccting the accuracy of the measurement of an; 

 /.cnith distance : Imt a very large error of colliin.-itioii would 1 

 venient in pi.ietiee. Tin' .-ec.ind pointer n, if exactly in a ill 

 with A, anil the di\ ' ly give the same result 



.1* \. This also may h.ive an error of colliniatioii. which may be treated 

 precisely as that of A ; or rather, the error of eollimatioii for tin 

 of two or more pointers is determined at once and by the sain 

 ration. Two readings have this advantage over one, that if some of 

 the divisions should ! srroneously placed, it is not likely that 

 errors in the same direction should fall at the same time under both 

 pointers ; and in any other supposition, the mean of the t v 

 will be affected by a loss error than one of them. Hence the advantage 

 of multiplying the readings for lessening the errors of division . 

 evident that what is improbable for two, will be all but impossible f..i 

 six, which is the number used in the Greenwich circles. There is. 



