LKSSONS IX ASTRONOMY. 



It. 



Assuming A = B = C in i i 



Sin. 3A = 3 sin. A cos. A - sin. 3 A 



= 3 in. A (1 - iu. ; A) - din." A 

 = 3 MIII. A - 3 BUI. S A - uiu. 3 A , 



.'. ain. 3A = 3 sin. A - 4 sin. J A (57) 



Similarly, from (50), OOH. 3A = 4 eos. 3 A - 3 co~ 



an.l from (51), tun. 3A ...(59) 



1 - 



\\ II. Formula /< tin- Ratios of an Any I e in terms of the 

 Ratios qf the Sub-multiples of that Angle. Sul.-tituting A for 



2A on tho loft-hand side of (52) to (56), and therefore 



- 



on tho right-hand side, we have 



Sin. A = 2 sin. A oos. A 

 2 2 



Cos. 



= oos. jA - sin. jA 



Cos. A = 2 cos. 2 f - 1 



a 



Cos. A = 1 - 2 sin. 2 A 



Tan.A= 2tan -* A . 



1 ;: ..A 



(60) 

 (61) 

 (62) 

 (63) 

 (64) 



From (57), (58), and (59), like formula may be obtained, by 

 like means, for sin. A, cos. A, tan. A, in terms of the same ratios 



A 



of - . Tho student should do this for himself. 

 3 



In this lesson have been given those, formulie most likely to 

 occur in after-practice. The student should not be content 

 with reading the demonstrations, but should in every case write 

 them out as he follows the proof, inserting any intermediate 

 steps which, from their simple character, may have boon omitted 

 to save space. He should also arrange new formulae for him- 

 self, as may be done to any extent by simple substitutions, or by 

 lidditions, subtractions, and divisions of formulae already given. 



KEY TO EXERCISES IN LESSONS 



NOMETRY. I. 



1. Sin. A = '6247. 2. Sin. A = "8930. 

 4. Sin. A = '8. 5. Cot. A = 2. 



6. Sin. A = '866 ; cos. A = '5 ; tan. A = 1732; cot. A = '5773; 

 sec. A = 2 ; cosec. A = 11547; covers. A ^ 131. 



. 1 _ gin ^ _ 1 - 8in. A = cos.* A . cos. A 



sin. A sin. A sin. A 



= cos. A . cot. A. - 

 . 1 J^coe. A _ 1 

 1 - cos.* A 1 - COB. A' 



7. 



IN PLANE TRIGO- 

 3. Cos. A = -9766. 



cos. A . 



LESSONS IN ASTRONOMY. XVIII. 



THE FIXED STARS : THEIR MAGNITUDES AND DISTANCES 

 SHAPE OF OUR CLUSTER DOUBLE STARS COLOURED 

 STARS VARIABLE STARS. 



WE must now turn our attention from the planets to the fixed 

 stars which so thickly stud the sky. It is very difficult by mere 

 inspection to form any estimate of the number of these bodies ; 

 it appears, however, from catalogues which have been compiled, 

 that the total number visible to the naked "ye is about 6,000. 

 Only half of the sky, however, can be seen at one time, and the 

 number visible on a clear night may therefore be set down 

 roughly at 3,000. These stars vary very greatly in brilliancy 

 and apparent size, and have accordingly been divided into six 

 classes, the brightest being said to be of the first maprnitudo, 

 while the faintest visible to tho naked eye are classed as the 

 sixth, the rest being divided into tho remaining four magnitudes. 



As a general rule, it is computed that stars of the first magni- 

 tude are about 100 times as brilliant as those of the sixth. The 

 light of Sirius, the brightest star in the sky, is, however, esti- 

 mated to be equal to that of 324 of the latter. 



Though tho number of stars seen by the naked eye is thus 

 limited, we must not suppose that these are all that exist. If 

 we direct a telescope to any part of the sky, we shall at once 



perceive that tho field of view U covered with point* of light, 

 and the number of these telescopic stars i* found to be im- 

 mennely greater than that of thoM vwible to the naked eje. 

 Thoo stars are classed into magnitude* down to the fifteenth 

 roenth, or even lower, according to the power of t ).. 

 telescope required to show them. The total number down to- 

 the font ti-i-.'it li magnitude a estimated at 20,000,000. 



The i|ii'--ti..n now suggest* itself whether thene different 

 degrees of brightness result from differences in the size of the 

 stars, or in their distances. To this we cannot give an answer 

 with absolute certainty, as there are only a few stars whose di* 

 have been manured. There appears, however, to be little 

 doubt that the difference is chiefly in their distances. The stars, 

 instead of being ranged in a cphere around us, as at first sight 

 they seem to be, are scattered through boundless space, and 

 placed at varying distances from us and from one anot! tr. They 

 are all likewise in motion round the centre of gravity of the 

 whole cluster. 



The distances of the stars are ascertained in the same manner 

 as those of the Sun and planets that is, by parallax. Instead, 

 however, of taking two stations at different parts of the Earth's 

 surface, and laying down a base line between them, we take the 

 diameter of the Earth's orbit, or 183,000,000 miles, as the base, 

 the observations being taken at intervals of six months. 



Even with this immense line, however, the parallax is so 

 small that it can only be detected by the most careful observa- 

 tions and accurate instruments. In no case has it been found te 

 be greater than 1" ; and if this be its value, the distance of the 

 star must be 206,000 times as great as that of the Sen. The 

 parallax of about a dozen stars has now been ascertained, and is 

 found to vary between 0'919'' and 0*046''. The star a Centauri 

 is the nearest to the Earth, and its distance is estimated at 

 20,496,000,000,000, or more than 20 billions of miles ; while the 

 average distance of stars of the first magnitude is probably 

 three or four times as great as this. These figures, however, 

 fail to convey to the mind any definite idea as to the real dis- 

 tance ; perhaps the best mode of expressing it is by stating that 

 light, with its speed of 184,000 miles a second, takes 3| years to 

 travel from that star to us ; while the smaller telescopic stars 

 are so remote that it must require upwards of 5,000 years for 

 their light to reach us. 



The rays by which we now see these stars must have left 

 them soon after the creation of Adam ; and, for aught we know, 

 some of them may for ages have ceased to exist. At the 

 contemplation of these things, however, the mind is altogether 

 lost in wonder ; we are verging on the infinite, and are led to 

 feel that this planet is indeed but a minute speck in the im- 

 mensity of creation. 



In studying the stars we need some mode of identifying them, 

 and in this there is some little difficulty. Special names have 

 been assigned to many of the more brilliant ones, but these have 

 a tendency to confuse. At a very early period they were divided 

 into constellations ; many new ones have since been added, so as 

 to make in all 109. Several of these, however, are very small 

 and unimportant, and hence are rejected by some astronomers. 



In 1604, a German astronomer, named Bayer, published a 

 celestial atlas in which he designated the stars in each constella- 

 tion by the letters of the Greek alphabet, the brightest being 

 called a, the next )3, and so on. This plan was found to answer 

 so well that it has been continued to the present time. In some 

 constellations, however, the number of stars now catalogued is 

 so great that more letters are required to denote them ; the 

 English alphabet therefore follows the Greek, and if both prove 

 insufficient, the remaining stars are denoted by numbers. 



In a few instances the stars are not arranged quite in the 

 ord^r of brightness, either from want of accuracy in Bayer's 

 observations, or from a change in the light of the star since 

 his time ; it is considered better, however, not to attempt to 

 amend this, as it would only produce confusion. 



The best plan for the student to become practically familiar 

 with the different constellations is to study the sky itself, with the 

 aid of some maps or of a globe. Several of the constellation.* 

 as, for instance, the Pleiades, the V-shaped cluster of the 

 Hyades, and Orion (Fig. 43), with the three stars in the belt, 

 commonly known as the Yard Measure are familiar to almost 

 every one ; these will serve as a guide in determining others. 



The stars are all of them bright, self-luminous bodies like our 

 Sun, which in all probability ap|>ear8 to other worlds to be one 



