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 x, so that means between two given numbers, did not place those <-1 (中 x) is x ; then ↓ (4−1x-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 4x=4(4x-1) gives pr.p'x= Fourthly, we have the short but interesting period in which, before the formal invention of fluxions or the dif-(pr). 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 with 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 ar tinguish between algebraical and transcendental problems. and (b) 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.

-1

x, the

As science advances, quantities which are now called transcendental will lose the name, and be received among 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 expression is fe x"-1dx taken from 0 to x=; and when n is an integer it is simply 1 × 2 × 3 ×....X". But when n is a fraction it can only be calculated by series. Nevertheless, as tables are now formed of its values, and as stands in as favourable a position for use as ordinary logamany properties and consequences of it are known, it

Sixthly, we have an alteration which it might have been supposed should have come long before, namely, the expression of the old transcendentals as recognised functions, and the writing of them accordingly, as log x, sin x, cos x, &c. Strange as it may appear, this was never formally and steadily done till the time of Euler. And it is only in our own day that the system has been completed by the recognition of the number whose logarithm is angle whose sign is x, &c. as functions of x, and the adop-rithms at the end of the seventeenth century. tion of the appropriate symbols log , sin1x, &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 18 algebraical when it is finite in form, and a 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 r 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 trans

cendental is not opposed to algebraical, but to that which is expressible by ordinary means, log r and sin a 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 pr.p'x = p(px), where p'r means the differential coefficient of or. 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 +/- 1. Let a, b, c, &c. be any of these roots, and let fr be a function of a formed as follows:4x = Aa”+ 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. Trans cendental 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 philo sophy is purely transcendental; while in the former it is ot a mixed character. [METAPHYSICS.]

T

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 hap pens 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 re marks 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 squam 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 a2 = 4 is satisfied by a = 2, or x=-2 because 2 x 2 = 4, and -2x-2=4; but how are we to say that there are no other values which satisfy this equa tion? When we transform the equation a=4 into (x-2) (x+2)=0, with which it is identical, we then see that this product can only vanish when 2 or a +2 vanishes; that is, when a is +2 or

2.

of

Transformations frequently leave a point unsettled which can only be determined by a subsequent species of expert mental test; or, lest the word experimental as applied to mathematical reasoning should give alarm, by a process 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

TRA

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 p+qɛ+ rx2 + ...., and by the use of the property above alluded to, it is found that the series must be of the form } + Ax + ‡ A2x2 + · ·.. But, as noticed in SYNTHESIS, all that is here proved is, that if a be capable of expansion in integer powers of x, 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 infend 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 conider 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.

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

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

x-p=x', y-v=y'.

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

Next, when the coordinates are those of points in space. The only two cases which are particularly useful are when both systems are rectangular, and when the new one only is oblique. Let x, y, z be the old coordinates, and 1,1,

the new ones. Let A, μ, v be the old coordinates of the new origin, and let the angle made by x and y be, that of y, and z be, and that of 21 and a1 be ŋ, which we may thus denote :—

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

By+B'y'+B"y"=cos &.
ya+y'a'+ya' = cos n
aß+a'ß'+a"B" = cos 2.

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, And first as to coordinates in one given plane. one set being in each column: the meanings of a, a', &c. 1. Both systems oblique. Let x and y be the old coorLet and v being as before, dinates of a point, x' and y' the new ones. μ 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 gle made by with a means the amount of revolution which would bring the positive part of x into the direction of the positive part of a, the revolution being made in the positive direction.

a2+a22+a2=1

By+B'y' +B"y"=0
ya+ y'a'+y"a"=0
aß+a'ß'+a"8"=0

a'a"+B'B"+7'7"=0 a"a+B"B+y"y=0 aa'+ BB' + vy' =0 VOL. XXV.-R

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:

desire to learn the origin of modern practical astronomy.

The object of the present article is to give such an account of the transit as will enable any one to use it with tolerable success. Those who wish for more perfect information must consult the introductions to the Greenwich, Königsburg, Dorpat, Cambridge, Edinburgh, &c. Observations. Our type will be the portable transit-instrument, leaving the reader to accommodate what is here said to the powers' of his own instrument, or to the practice of the observatory which he adopts for a model.

For the mode in which these nine quantities are made to depend upon three, we must refer to works on mechanics, in which such reduction is particularly useful. We avoid giving it here, because trifling differences exist in the manner of taking the quantities to functions of which all the rest are to be reduced, so that no set of equations can be given which can be called universal. As far as we have gone, the expressions of all writers are the same, though the letters used are not always alike. TRANSFUSION OF BLOOD is the operation of trans-be of great strength, though perhaps that which is here ferring the blood of one animal into the blood-vessels of another, and is sometimes beneficially employed for reviving those who are nearly dying after severe hæmorrhage. The operation had long been used as a means of experiment, and in the vain hope that by injecting the blood of a healthy man or animal into the vessels of a diseased one, the health of the latter would be restored; but it had rarely been employed for its only useful purpose, till Dr. Blundell, after a long series of well-conducted experiments on animals, proved that it might be safely and advantageously employed in men. His observations are published in his Physiological and Pathological Researches ;' and since his revival of the operation, the lives of many persons have been saved who were, in all probability, dying from the loss of blood during or after surgical operations, during gestation, and in other circumstances. The operation has indeed often failed; it has often been unnecessarily performed; and its performance is not unaccompanied by danger to the patient; but still there is sufficient evidence of its high utility in cases which, without it, would have been quite or nearly hopeless, to warrant its being resorted to under the guidance of a sound judgment.

There are three principal parts expressed in the plate. The iron stand, carrying the Y's with their adjustments; the telescope, inserted at right angles through an axis with a small vertical circle for finding or verifying stars; and the cross level. The stand is made of cast-iron, and should figured would be found inconveniently heavy if the instru ment is often moved. The Y's are contained in brass pieces, strongly united to the tops of the two uprights. The left hand Y has a motion up and down, which is given by a milled screw partially seen immediately under the pivot. The right hand Y is moved in azimuth by a screw, the milled head of which is seen projected upon the lantern. In portable instruments it is very convenient to have this lateral or azimuthal adjustment made by screwing against a spring, as it is in this instrument. In fixed observatories the adjustment is made by two antagonist drawing screws, one of which is tightened and the other loosened; and indeed this is the general construction of instruments of every size, and is the most solid fixture. But it is so convenient to be able to move the instrument at pleasure in azimuth while actually looking through the telescope, that we should strongly recommend the adoption of the counterspring whenever the instrument is small, and is either to be frequently shifted, or is not furnished with a meridian mark. The spring must press pretty strongly against the adjustment, to keep all secure. screw, and there should be a clamping button in each

The axis is made of two strong brass cones soldered on The chief instruments employed in the operation are a the central sphere. syringe, with double pipes, a basin of, appropriate form, shoulders, over which the cones slip. As this is the The sphere is cast hollow with two and a fine tube fixed on one of the pipes of the syringe. most important part of the instrument, great care should One of the veins of the arm of the patient being opened be taken of the fitting before the axis is finally put tojust sufficiently to admit the point of the tube, and fixed gether, and the symmetry of the parts as to the centre by a probe, blood must be drawn through a free opening should be perfect. If the instrument is weak here, it is in the vein of some healthy person, and as it flows into the utterly worthless. In the older English instruments the basin must be slowly sucked up, without any mixture of centre was a cube, and that form is frequently adopted at air, by the syringe. When the syringe is filled and care- present by continental artists. The transit at Bruxelles fully cleared from all air by forcing blood up to the very by Gambey, one of the largest and finest instruments in point of the tube, the latter must be introduced into the the world, is so patient's vein, and the blood steadily and slowly injected. however is symmetrical strength, and any shape is good constructed. The essential requisite Four or five ounces are often sufficient to revive a patient, which fulfils this condition. The pivots are soldered into and if they produce head-ache, flushings of the face, the extremities of the cones, and are turned after the whole tendency to fainting, and other unpleasant symptoms, the is fixed. One of them is pierced to admit light into the transfusion should be arrested; but if not, the injection axis. should be continued till it produces some good effect, or face of steel, which is less affected by wear. Greater care In large instruments the pivots have an outer surtill a pint of blood has been transfused. Beyond this it is is required to guard steel pivots from rust,* and the turning not safe to carry the operation, nor is it likely to be must be performed with a diamond cutter, as the hand beneficial. A second or a third injection may be em-knots to which steel is subject resist and jar the ordinary ployed when the state of the patient seems to render it

necessary.

The experiments of transfusing the blood of various animals into the vessels of man proved only mischievous; and those of transferring the blood of an animal of one species to the blood of another species are of too little interest and have produced too few general results to be worth recording here. The injection of various medicinal

substances into the veins has been tried, but its effects are not sufficiently different from those produced by the ordinary mode of taking medicine, to render it advisable to submit to an operation which is itself dangerous. All the important facts relating to the subject may be read in an article on Transfusion, by Dr. Kay, in the Cyclopædia of Practical Medicine,' and in the works from which he

quotes.

TRANSIT, or TRANSIT INSTRUMENT (Instrument des Passages), was invented by Römer about the year 1690. The description is to be found at page 47 of the Basis Astronomia,' by his pupil Horrebow, Havniæ, 1735; and we recommend the perusal of this book, which contains an account of Römer's inventions and methods,

cutter out of its place. The pivots should be turned pretty
nearly to the same diameter: the marks of the tool are
ground off afterwards by collars which are made to fit closely
on the pivots, and are changed and reversed from time to
time.
should be discontinued, as a small difference of size in the
When the surface is perfectly formed, the grinding
pivots is of little consequence, while an alteration of the
cylindrical form of the pivots, or of the direction of their
end of the axis is on the right-hand pier in this figure.
axes, ruins the instrument. The perforated or illuminated
The light of the lantern shines through this and lights up
of 45° with the axis and with the telescope, and thus light
an annular plate in the centre, which makes an angle
enough is thrown down to the eye-end to illuminate the
field very vividly, while the opening allows the rave
quantity of light may be regulated by a contrivance for
from the object-end to pass without impediment. The
diminishing the aperture of the lantern, or by a shade
passing between the lantern and the pivot. In some
* When the instrument is small and frequently out of use, it should be re
them.
moved from the Y's, and the pivots protected by caps which slip closely over

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