had brought into use sines, cosines, &c., and logarithms, where one, two, or any number of roots may be taken at which were then properly transcendental. The words pleasure: and A, B, &c. are any quantities independent of which described a particular mode of drawing lines in a circle, or the result of many interpositions of geometrical. Let be the inverse function of 4, that means between two given numbers, did not place those (4x) is x; then (x-1) is a solution of the on lines or means among the objects of algebra, and gave no clue to any algebraical properties.

ginal equation, or pr(x-1) gives px.px= Fourthly, we have the short but interesting period in which, before the formal invention of fluxions or the dif- (px). Now is, when more than one root is used, ferential calculus, infinite series began to be employed, and inexpressible except by infinite series: that is, not merely the transcendentals last alluded to ceased to be absolutely inexpressible in common algebraical terms, but even wit incapable of expression. This was the state in which the assistance of logarithms and trigonometrical functions Leibnitz found the science when he first proposed to dis-Nevertheless, as particular cases of this solution, both ax tinguish between algebraical and transcendental problems. and (b2-x) are found. Fifthly, we have the period succeeding the invention of the differential calculus, in which the areas and lengths &c. of curves could be expressed, whether they could be reduced into older language or not, by the new signs for fluents or integrals.

Sixthly, we have an alteration which it might have been

As science advances, quantities which are now called transcendental will lose the name, and be received amon the ordinary modes of expression of analysis. One of the first of these will be the well-known function of n, which is generally designated by rn, and is sometimes called the gamma-function, sometimes the factorial function. Its

supposed should have come long before, namely, the ex-expression is ƒ € x-1dx taken from x = 0 to x = ∞; pression of the old transcendentals as recognised functions, and when n is an integer it is simply 1 × 2 × 3 ×....Xn. and the writing of them accordingly, as log x, sin x, cos x, But when n is a fraction it can only be calculated by series. &c. Strange as it may appear, this was never formally and steadily done till the time of Euler. And it is only in Nevertheless, as tables are now formed of its values, and as our own day that the system has been completed by the many properties and consequences of it are known, it recognition of the number whose logarithm is x, the stands in as favourable a position for use as ordinary logarithms at the end of the seventeenth century. angle whose sign is x, &c. as functions of x, and the adoption of the appropriate symbols log x, sin-1x, &c.

Seventhly, a most important addition has been coming into use in the present century, namely, the employment of definite integrals as modes of expression, not merely of functions of the variable of integration, but of other quantities which only enter as constants, or which, if they vary, vary independently of the variable used in integration. So powerful is this mode of expression, that it may almost be suspected to be final and the word transcendental is rapidly acquiring a new meaning. We predict that it will settle into the following: a transcendental result will be one which is incapable of expression except by a definite integral, or by an infinite series which cannot be otherwise expressed than by a definite integral.

In the meanwhile there are two senses in which the word is used. The first is that just explained; the second has reference to the old distinction of algebraical and transcendental. A function of x is algebraical when it is finite in form, and x is never seen, nor any function of it, in an exponent, nor under the symbols of a sine, cosine, &c., or a logarithm. No operation then enters with a unless it be one of the four great operations of arithmetic, or else involution or evolution with a definite exponent. Thus in this sense of the word, log x and sin x are both transcendentals. But in the modern sense in which transcendental is not opposed to algebraical, but to that which is expressible by ordinary means, log x and sin x are not transcendental, being among the most common of the present modes of expression, and being, in fact, connected with algebra in a way which, had it been understood when these symbols were first used, would probably have always saved them from the distinctive term.

The roots of equations of the fifth and higher degrees are, properly speaking, transcendental: there is no mode of expression except by infinite series. And, generally speaking, and with the exception of a few cases in which modes of expression have been invented and studied, INVERSE functions are transcendental. And a result of such inversions, even though, from our ignorance of its real properties, it may be expressible by ordinary means, is transcendental as long as that ignorance lasts. And it is useful to observe that forms of the most different kind may be connected together by such a relation as this, that both are cases contained under the same transcendental.

To exhibit the arrival of one of these transcendentals of inversion, as they might be called, let us take the equation px.p'x = p(px), where p'r means the differential coefficient of px. A large class of solutions may be obtained as follows:-The equation y log y = C has an infinite number of roots, two at most being real, and all the rest of the form a + B-1. Let a, b, c, &c. be any of these roots, and let x be a function of a formed as follows:4x = Aaˆ+ Bb®+.........

TRANSCENDENTAL, a technical term in philosophy, derived from the Latin transcendere, to go beyond a certain boundary. In philosophy transcendental signifies anything which lies beyond the bounds of our experience, or which does not come within the reach of our senses. It is thus opposed to empirical, which may be applied to all things which lie within our experience. All philosophy therefore which carries its investigations beyond the sphere of things that fall under our senses is transcendental, and the term is thus synonymous with metaphysical. Transcendental philosophy may begin with experience, and thence proceed beyond it; or it may start from ideas à priori which are in our mind: in the latter case philosophy is purely transcendental; while in the former it is of a mixed character. [METAPHYSICS.]

TRANSFORMATION, a general term of mathematics, indicating a change made in the object of a problem or the shape of a formula, in such manner that the original problem or formula is more easily solved, calculated, or used after the transformation. Thus it frequently happens that the solution of an equation is facilitated by reducing it to another equation having roots which bear simply a relation to the roots of the former: as an instance, we may refer to the solution of the cubic equation in the article IRREDUCIBLE CASE.

All the process of algebra consists in transformation, from and after the point at which the problem to be solved is reduced to an equation: so that to write on this subject in detail would require an article on algebra. A few remarks on the leading points which present themselves in transformations are all we can here undertake to give.

It frequently happens that transformation points out the nature of a consequence in a manner by which the direct reasoning of algebra is strongly confirmed and illustrated. For instance, when we assert that a quantity has two square roots, one positive and one negative, our assertion is easily verified in its positive part: but it does not follow by the same reasoning that a quantity has only two square roots. We may say that a 4 is satisfied by x= 2, or x=-2 because 2 × 2 = 4, and −2 × −2 = 4; but how are we to say that there are no other values which satisfy this equation? When we transform the equation_x=4 into (x-2) (x + 2) = 0, with which it is identical, we then sec that this product can only vanish when x - - 2 or x + 2 vanishes; that is, when x is +2 or — 2.

Transformations frequently leave a point unsettled which can only be determined by a subsequent species of expermental test; or, lest the word experimental as applied to mathematical reasoning should give alarm, by a process of detection which is to choose between alternatives which the process of transformation leaves undecided. This fre quently happens when the nature of the transformation is ascertained by means not of the expression to be trans formed, but of one of its particular properties. For in

x-μ=cos p.x2-sin p.y'
y-v=sin.x' + cos p.y'

5. The coordinates of the new system parallel to those Here of the old one.

x-μ=x', y-v=y'.

In any of the preceding cases, if the new and old origin coincide, we have only to make μ=0, v=0, and use the formulæ accordingly.

new origin, and let the angle made by x, and y1 be %, that of y and z1 be , and that of zi and i be n, which we

may

thus denote :

This brings us to the mention of a defect of reasoning which has frequently vitiated mathematical works, namely, the assumption of the species of a transformation, and the supposition that only the character of the details remains to be settled, or the individual of the species to be picked out. In the preceding case, for example, it is often stated as follows:-Required the expansion of a in a series of powers of x.' The form of the series is then assumed, say Next, when the coordinates are those of points in space. p+qx+rx2 + . ..., and by the use of the property above The only two cases which are particularly useful are when alluded to, it is found that the series must be of the form both systems are rectangular, and when the new one only 1 + Ax + † Â2x2+.... But, as noticed in SYNTHESIS, all is oblique. Let x, y, z be the old coordinates, and 1, y1, that is here proved is, that if a be capable of expansion, the new ones. Let λ, p, v be the old coordinates of the in integer powers of r, the expansion must be of the form 1+ Ax + ...... It is true that, looking at what we see in algebra, that science might be strongly suspected to have a peculiar power of rejecting false suppositions, or of indicating their falsehood by refusing to furnish rational results: thus it certainly does generally happen that when we attempt to select from among series of integer powers the one belonging to an expression which really has no such series, we find infinite coefficients, or some other warning. But it is too much to ask of a beginner that he should take it for granted that algebra has so peculiar a property; nor, in fact, is it true that such a property is quite universal. It is necessary therefore to watch all transformations narrowly, both in their general as well as their specific form: first, because there can be no sound reasoning without such caution; next, because, though it is true that in many parts of algebra the science will refuse to acknowledge and obey a false assumption of form, yet it is almost impossible to draw the line at which this refusal ends, and the idea that such a power is universal in algebra will lead the student to many a serious difficulty in the higher branches of mathematics.

TRANSFORMATION OF COORDINATES. We intend this article purely for reference: that is, supposing the subject already known, we mean only to put together the formulæ in such a manner that any one can be used at once.

Rectilinear coordinates are the only ones which are usually transformed; such a thing rarely, if ever, happens with polar coordinates, except in investigations each of which has its peculiar method. And first we shall consider rectilinear coordinates in one plane, and afterwards in space. What is usually wanted' is to express the coordinates of a first system in terms of those of a second, and subsequently given, system.

And first as to coordinates in one given plane.

1. Both systems oblique. Let x and y be the old coordinates of a point, x' and y' the new ones. Let μ and be the old coordinates of the new origin: 0 the angle made by the old coordinates: the angle made by the axis of

with : the angle made by y' with x. Angles are to be measured as explained in the article SIGN: thus the angle made by x' with x means the amount of revolution which would bring the positive part of x into the direction of the positive part of ', the revolution being made in the positive direction.

=

Then we have the following formulæ :—
x-λ= ax1 +ẞy1+yzı
y-p=a'x1 +ẞ'yı+y'zı
2-v=a"x1+ẞ"y1+y"zı.

Where the meanings of a, ß, &c., and the connection of those meanings with the places of the letters in the formulæ, will be easily caught from the following:

a = cos xx1, ẞ= cos xy1, y = cos x 21

^

^

a' = cos y x1, ẞ'= cos y y1 y' = cos y z1

Λ

a"= cos zx1, ẞ"= cos zy1 y": =COS 2 21. And a, a', &c. are subject to the following six conditions:

This case is not much required: the following, in which both systems are rectangular, is of the highest importance. When we speak of the angle made by two axes, we mean, as before, the angle made by the positive side of one with that of the other; but, since only cosines are used, the direction of revolution is immaterial. If both systems be rectangular, and if they have the same origin, we have two sets of equations, each of which follows from the other, one set being in each column: the meanings of a, a', &c. being as before,

Besides which, each of the quantities a, a', &c. may be | to all those who, reading Latin with moderate ease, feel a expressed in terms of the others, as follows:

looked at, two out of three small screws are seen which | nect the telescope with the axis. We are not disposed to fix the circle to the axis. When these are released, the attach much value to this mode of connecting the axis and circle will turn freely round. This contrivance will save telescope, which moreover requires very accurate fittings. some trouble when an instrument is used for a long time The braces are positively injurious, unless they are exactly in the same place without reversing, but is scarcely worth and at the same moment exposed to the same temperature. being applied to one which is frequently shifted or reversed. It is said indeed, in the memoir just mentioned, that when The clamp for fixing the telescope in altitude and the the antagonist braces were exposed to very different temslow-motion screw are seen at b. There is a caution to peratures, the instrument continued to preserve its form. be given here. The tail-piece should never be tightly If so, the experiments simply show the centre-work to have nipped, unless the instrument is used for observing declina- been so strong that the braces could not disturb it, in which tions, and it and the tangent screw must be released case they are merely useless. At Cambridge the braces were when the observer uses the azimuth screw for bisecting found to derange the instrument, and were consequently any object, such as a mark or a star at a given mo- removed, to the great improvement of its steadiness. ment. In large transits there are generally two small There is no great difference of construction between circles fixed on each side of the transit towards the eye-end. different transits, except what we have already mentioned. They are here more convenient for setting, and it is It is desirable even for the smallest instruments that the easy to pass rapidly from one star to another, when both supports should be of stone when they are not perpetually the circles are previously set. There is great diversity in shifted about. The Y's then are separate pieces fixed by the graduation of the setting circles. In large instruments screws to plugs let into the stone. For small transits the which are used for some time in the same position, it is stone may be in one or three pieces, according to the size. best to make the verniers read polar distance or declina- When practicable, the piers should be high enough and tion, re-adjusting the circles whenever the transit is wide enough apart to let the observer stand or lie down reversed. With a portable transit, which is or ought to between them. This saves perpetual meddling with the be very frequently reversed, a graduation to altitudes one eye-piece and the eye is less strained. We have already way, which becomes zenith distances when reversed, is per- remarked that the performance of a well-made transit dehaps as convenient as any, though a slight computation pends rather on the permanence of its fixing than anything for each star is required to form a working catalogue. else. It is to the greater care bestowed on the foundations The telescope in this instrument is not inserted in the ordi- of large instruments that much of their superior performnary manner. The central portion, from c to d, is in one ance is to be attributed. tube, pierced on the right side to allow the light to pass, and soldered at e and f to the central sphere. The reflecting plate is fixed in this telescope, and can be turned to throw the light up or down. The object-end and eye-end are screwed on at c and d, and are interchangeable. The telescopes are usually in two pieces, which are screwed into the central sphere at e and f. The advantage expected from the present construction is, that there is firmer screwhold and less leverage for any blow or rough handling; and that by interchanging the object and eye end, fresh portions of the pivots are brought into action, thus diminishing wear and equalizing minute errors of form or flexure. The object-glass of the telescope should be carefully selected, and of as large an aperture as will show a good image. The superiority of a large instrument over a small one is wholly in the increased optical power. In all other respects it is probably inferior, .e. if the support of the smaller instrument be as solid as that of the larger. There are seven fixed vertical wires at equal spaces, and two horizontal wires, between which the star is observed. The head of the micrometer is shown at g. A small prism for observing stars near the zenith is slipped on the eye-piece when required, as at h.

The level rides on the pivots with its Y's. There is a pin at each end, which drops into a fork at i, to hold the level safely and upright. This is completely seen at the left pier. At this end is the adjustment for setting the level tube parallel with the axis. At the other end is an adjustment for raising or depressing that extremity of the level. The level should be very sensible and of the same curvature throughout. The graduation we have found most convenient is to have the principal divisions to 15" and the subdivision to 1"-5, numbered as units and tenths, which, though erroneously, is briefly described by calling the units seconds of time. If this scale should be too fine for the level, a principal division to 30" and subdivisions to 3", but still numbered as units and tenths, will be found equally convenient. The riding level is generally applied to the instruments which are so large, and consequently the piers so high, that a man cannot apply the level safely while standing on the floor, and also to small instruments, of necessity, when they are clamped, as this is, to the pier. For a transit between stone piers which does not exceed five or six feet, we prefer a swinging level, which may be applied and read while standing on the floor.

When Troughton undertook, much against his will, to construct a ten-foot transit for the Royal Observatory, he adopted a very ingenious mode of uniting the cones and the telescope with the central sphere. The description will be found in the Phil. Trans.' for 1826, p. 423: that part which treats of the construction of the instrument is from Troughton's own pen. He also added four braces, to con

The principal use of a transit instrument is that of determining the exact moment when a celestial body passes the meridian of the place of observation. Now the meridian is a great circle which passes through the zenith and the pole, and the instrument is adjusted when the line of sight is a portion of the meridian during the whole rotation of the telescope.

As in all other instruments, the telescope is first to be adjusted for distinct vision. Put on a tolerably high power, and slide the eye-piece out and in till you see the wires sharply without straining the eye. Then direct the telescope to a bright star or a double star; and if the image of it is distinct, the telescope is in focus. If not, release the screws at k, and draw the tube out or push it in until the image is as perfect as you can make it. There is another opposite screw to k, and the exterior holes allow a little play. Some trouble and guesswork may be saved by making two slight scratches on the eye-piece where the sight of the wires and of the star are respectively most perfect, and drawing the principal tube out or pushing it in this quantity. The operation has succeeded, if, in viewing a slow-moving star, like Polaris, there is no shifting between the star and the wire which bisects it, while the observer moves his head laterally. This adjustment is generally best made by the instrument-maker, and as it is not liable to alter, we should prefer to have the telescope tube cut the proper length upon his responsibility, so that the position of the wire is permanent. When this adjustment is completed, the telescope must be turned on some tolerably distinct object, which is to be bisected by the middle wire near the upper part of the field: if on raising the telescope it is also bisected at the lower part of the field, the wire is perpendicular to the axis; but if not, the tube is to be twisted without altering the focal length until the object comes half way to bisection: the bisection is completed by the azimuth-screw, when the object ought to be seen bisected at the top of the field when the telescope is depressed. One or two trials will suffice for this purpose, and then the screws at k must be tightened.

The first of the principal adjustments is that of setting the line of sight at right angles to the cross-axis, when it necessarily describes a great circle. A distinct object must be selected not far from the horizon, and bisected by the middle wire, using the azimuth screw. The axis is then carefully lifted out of its Y's, and returned end for end, or reversed, and the object viewed again. It is now to be bisected as before, half by moving the azımuth-screw, and half by the screw at 7 and its antagonist, each of which draws the plate on which the wires are fixed. The operation must be repeated until no dif

We shall speak at present as if there were only one fixed wire in the telescope, viz. the middle vertical wire. The subsequent modification will cause no difficulty.