VAREC. 



VARIABLE STARS. 



654 



an abrupt character, and productive of as many vapour-planes ; corre- 

 sponding interruptions in the decreasing progression of temperature 

 are sometimes distinguishable, but in a less degree, as might indeed be 

 expected from the fact that at greater elevations, and consequently 

 lower temperatures, the variations in the absolute amount of aqueous 

 vapour are necessarily smaller, and their thermic effects consequently 

 diminished. 



VAREC. An obsolete name for crude carbonate of soda. 



VARIABLE. A quantity is said to vary when it changes value, 

 whether gradually, or by jumps or starts. The notion of a variable 

 quantity is the first which must be established in teaching the Diffe- 

 rential Calculus, and requires a little explanation. 



One magnitude at least is hardly conceivable without the notion 

 of variation ; we mean time or duration. Reckoning from a fixed 

 epoch, the idea of the present time is nothing but that of the other 

 extremity of a variable quantity, the variation of which we cannot 

 suspend, even in thought. Again, in space-magnitudes, though we are 

 not obliged to consider them as formed by variation, yet it is in em- 

 power to do so, and we are constantly learning the variation of length, 

 area, or solidity consequent upon motion. And we can even consider 

 this variation as arising from no act of our own, as independent of us, 

 and out of our power to stop : though even when this is physically 

 true, namely, that the variation is out of our power, we can conceive or 

 imagine that it does stop, and trace the consequences of such stoppage. 

 Variable magnitude, then, presents natural ideas, such as we not only 

 easily acquire, but such as it would be difficult, if not impossible, to 

 suppose that we could help acquiring. 



But when we come to speak of number, the case is much altered. 

 The constant phrase of an algebraist, " let x be a variable quantity," 

 clear as it may be when quantity means magnitude, is not quite so 

 plain when quantity means number as the representative of magnitude. 

 There is something to be said as to how number is imagined to vary 

 at all : and still more as to its gradual variation. 



Number is an abstraction of the mind ; it la not magnitude, but a 

 mode of reference of one magnitude to another. If we might dare to 

 say it, number is more of the nature of an opinion about magnitude 

 tli m of magnitude itself. When we speak of a symbol representing a 

 variable number, we know that, though we say the symbol changes its 

 value', it is we ourselves who arbitrarily change the meaning of the 

 symbol. We can imagine (waving all question about the possibility of 

 our imagination, or its metaphysical truth) everything annihilated 

 except two material points, one or both of which are in motion with 

 respect to the other : but we cannot in such a case imagine x to be a 

 symbol of a variable number. Unless some intellect be in existence to 

 mean something by x, or to make a symbol of x, there can be no such 

 thing as a variable number, or as the abstract idea of number at 

 all. When we say, let x be a variable number, we must always be 

 understood to mean, let x be a symbol which at one time we may be 

 allowed to make to stand for one number, and at another time for 

 another. 



Mow as to gradual variation. A point never changes its distance 

 fr. .m another by, say a font, without making every assignable lesser 

 change in the interval. Or, a line which is lengthened from A B to A c 

 by the motion of a point, must at some period of the change be equal 

 to A D, if A D be anything between A B and A 0. At least it is a neces- 

 nary condition of our existence to believe this to be as evident as that 

 two straight lines cannot inclose a space, though [Sr ACE AND TIM i: J 

 we believe some would be found to deny it. But in the case of number, 

 we cannot form anything but an approximation to this idea of gradual 

 variation. We can pass from 1 to 2 by successive steps, by millions of 

 millions of steps if we please : that is, h. representing a small fraction, 

 we can proceed from 1 to 2 by the steps 1 + A, 1 + 2A, 1 + 3A , &c., in 

 such manner that we shall not arrive at 2 till a million of million of 

 steps have been made. But this is not gradual variation, such as U 

 in our ideas when we think of a line increasing in length by the reces- 

 sion of one extremity from the other. Nor, if we subdivide our steps 

 ever so far, can we, in counting, cease to make steps ; that is, we 

 cannot imagine gradual variation of number. When, therefore, we 

 talk of x standing for a number, which is also to represent the number 

 of units in a variable length, we can only mean that our numerical 

 pn .greswion can be made, if we please, by steps so small, that whatever 

 length A D may represent, the linear representatives of some or other of 

 the numerical steps by which we pass from the number in A B to the 

 number in A c, may be made as near to A D as we please. It is, no 

 doubt, in this essential distinction between the ideas involved in the 

 variation of number and in that of magnitude, that the existence of 

 I -.< OMHENSURABLE quantities takes ita rise. 



The first steps of the Differential Calculus are often embarrassed by 

 a mode of speaking which appears as if two different symbols were 

 used for the same thing. " Thus," it is said, " let .c be a variable, and 

 y a function of that variable, such that y is always = a:'. Then let x 

 be changed into x + h, in consequence of which y becomes y + k; so 

 that y + k-(x + h)'." Now if x be the symbol of the variable quantity, 

 which can only mean this, that both before the quantity has changed, 

 and after, it is represented by x, how can it be allowed both to let x, 

 ta it were, imply its own variation in its very meaning and yet alter 

 x into x + A to denote that x changes ? The truth in that the language 

 U incorrect; it should bo as follows : - Let thuio IKI two variable 



quantities, one of which is always the square of the other ; let x be the 

 value first given to one of the variables, and y to the other, so that 

 y = x-. Then let a new value x + h be given to the first variable, in 

 consequence of which the second becomes y+k, so that f+k**(x+li)*. 

 In fact, a; does not represent a variable quantity, but a certain valuo 

 given to a variable quantity. 



VARIABLE STARS. This term has been applied to a class of 

 stars which exhibit variations of brightness when observed from time to 

 time. The branch of astronomy which takes cognisance of such objects 

 is entirely of modern origin. The first star of which the light wan 

 found to be variable is a small star in the constellation of the Whale, 

 usually designated in the catalogues of astronomers as o Ceti, or 

 omicron Ceti. This star was observed by Daniel Fabricius, on the 

 13th of August, 1596, and noted by him as a star of the third magnitude. 

 In the month of October of the same year, the stir had vanished from 

 observation, as if it had been extinguished, and for some time after- 

 wards it does not seem to have attracted the notice of observers. 

 Bayer, in his ' Uranomrtria,' published in 1603, has inserted the star, 

 but he makes no allusion to its previous disappearance. The discovery 

 of the variability of its light is due to Holwarda, a Dutch astronomer. 

 In the month of December, 1638, Holwarda perceived the star during 

 an eclipse of the moon, when it exceeded in brightness a star of the 

 third magnitude. About the middle of the following summer he was 

 unable to discover the slightest trace of it. However, on the 7th of 

 November, 1639, he again detected it in its original position. 



The star was now carefully observed by several individuals, among 

 the rest, by the famous Hevelius. The discovery of the period of its 

 variations is due to Bouillaud, who found that an interval of 333 days 

 elapsed between two successive disappearances or reappearances. Tho 

 results of modern observation indicate the exact period to be 331 a 

 15" 7". 



In addition to the star to which we have just been referring a great 

 many other stars have been found to be variable, and the number is 

 rapidly increasing every year. This important result is due in a great 

 degree to the practice of carefully scrutinising small stars in connection 

 with the search for asteroids which has been so assiduously prosecuted 

 in recent years by a number of individuals in different countries. We 

 shall now allude briefly to the peculiarities of two or three other ex- 

 amples of variable stars. 



;3 Persei. This star, usually termed Algol, which is situate in thu 

 head of Medusa, was found by Montanari and Maraldi to exhibit 

 strange fluctuations of brightness, but the period of its variations was 

 first established by Goodricke in 1782. It generally appears as a star 

 of the second magnitude. In the short space of three hours and a half 

 it descends to the fourth magnitude, and then in an equal interval of 

 time regains its usual brightness. It shines as a star of the second 

 magnitude during the space of two days; thirteen hours, and three- 

 quarters, and it consequently passes through the complete cycle of its 

 changes in two days, twenty hours, and three quarters. According to 

 Argelander, the exact period is 2* 20 h 48 52'. 



/3 Lyrte. This star was first found to be variable by Goodricke in 

 1784. It U an object of great interest, inasmuch as it possesses a 

 double maximum and a double minimum. When it arrives at its 

 maximum brightness, it resembles a star of the third magnitude. At 

 one of ita minima it appears between the third and fourth magnitude, 

 and at the other, between the fourth and fifth magnitude. Argelander 

 has found that it passes through its variations in 12 d 21 h 53'" 10*. 



8 Ccpliei. This interesting star was also first discovered to be variable 

 by Goodricke in 1784. Argelander determined its period to be 5 rt 8 h 

 47 39'6' ; but the late Mr. Johnson, director of the Radcliffe Obser- 

 vatory, Oxford, fixed the period at 5 J 6 h 42 18'4*. At its minimum 

 it is equal to a star of the fifth magnitude, and it hence increases until 

 it resembles a star between the third and fourth magnitude at its 

 maximum. The interval which elapses between the maximum and the 

 minimum is 3 d 19', while between the minimum and the maximum 

 the interval is only l d 14 h . 



Some stars have been discovered to be variable, but their fluctuations 

 are so irregular that it has been hitherto found impossible to reduce 

 them to any fixed law. A remarkable example of this kind is furnished 

 by the bright star in the southern hemisphere, denominated >j Argua. 

 In 1677, Halley, during his residence at St. Helena, classed it among 

 the stars of the fourth magnitude. In 1751, Lacaille estimated it to 

 be of the second magnitude ; however Burchell, who resided in South 

 Africa from 1811 to 1815, again ranked it among the stars of thu 

 second magnitude. From 1822 to 1826, it was estimated to be of the 

 second magnitude by Brisbane and Fallows, who observed it, the former 

 at New South Wales, and the latter at the Cape of Good Hope. In 

 1827, Burchell, while residing at St Paul's, Brazil, estimated it to be 

 of the first magnitude, and almost equal to o Crucis; but in tho 

 following year he found from observations made at Goyer, that it had 

 again descended to the second magnitude. Johnson, who observed tho 

 star at St. Helena between 1829 and 1833, estimates it to be of the 

 second magnitude; and Taylor's observations at Madras during the 

 same period indicate the same fact. Sir John Herschel also, from the 

 time of his arrival at the Cape of Good Hope in 1834 till 1837, esti- 

 mated it invariably to be between the first and second magnitude. 

 But on the 16th of December, 1837, while engaged in making photo- 

 metric uUiurvatioud of tho email stars in its vicinity, ho w.w mirjiri-'ud 



