coal-pits, and on the temperature of hot-springs. Mr. | been made by members of the Royal Society, and the numHopkins's Researches on Physical Geology' appear in the ber of papers relating to this subject is considerable. volume for 1840. It is remarkable that most of the wild Among them are accounts of the experiments made by the theories which have been formed to explain the cause of earliest electricians, Hauksbee, Gray, Dufay, and Desathe Noachian deluge and of the present state of the earth guliers. Dr. Watson has a paper (1748) on the distances are to be found in the volumes of the Transactions.' to which shocks may be conveyed by conductors, and anAmong the papers relating to Geography and Topo- other (1751) in which an account is given of Franklin's graphy is one by Mr. Murdoch (1758), describing the Treatise on Electricity. There is a paper by the Abbé method of representing part of the earth's surface on that Nollet (1748) on the effect produced by electricity on of a cone; but the most important papers belonging to water flowing through capillary tubes; one by Mr. Wilson these subjects are those which contain Dr. Maskelyne's (1759) on the heated tourmalin; and another by the same observations (1775) on the attraction of mountains, and (1773) on the efficacy of blunt conductors: there are the proceedings relating to the trigonometrical survey of also several papers by Dr. Priestley on electrical subjects; the British Isles by Generals Roy and Mudge and Colonel Mr. Cavendish's theory of electricity (1771): Volta's acColby. count (1800) of the galvanic pile; and Sir Humphry Davy's paper containing his great discovery concerning the agency of galvanism in decomposing compound substances; besides the highly valuable papers by Mr. Faraday concerning his experimental researches in electricity. Among the papers on magnetism are that of Dr. Brook Taylor (1715) on the law of magnetic attraction; the hypothesis of Halley concerning the cause of the dip and variation of the needle; and Lt.-Col. Sabine's recent contributions on terrestrial magnetism. The second great division is that of Mathematics, and on this important subject the papers are numerous. Among those which relate to the antient Geometry are the papers of Dr. Simson (1723), and of Mr. Brougham (1798), on Porisms; a problem by Dr. Pemberton (1763), concerning loci, and a paper by Dr. Horsley (1772), on the invention of Eratosthenes for determining prime numbers. There are nine papers on the origin of the nine digits; in one of which, by Mr. Barlow (1741), it is shown, from a date in the parish church of Romney, that the Arabic numerals must have been known in England about the year 1000. Of the rest, the most important are the logarithmotechnica of Mercator; the papers of Dr. Halley, Mr Cotes, and Mr. Hellins, which relate also to logarithms; the papers on the nature of equations, on series, and on annuities, also those which relate to the conic sections and to the quadrature of curves. In the later volumes will be found the important papers which have been contributed by the eminent mathematicians of the present day. Under the third great division, which is that of Mechanical Philosophy, are classed Astronomy, Optics, Dynamics, Mechanics, Hydrodynamics, Acoustics, Navigation, Electricity, and Magnetism; and among the papers relating to astronomy are those of Dr. Halley (1691, 1716), on the conjunctions of the inferior planets, and on the transits of Venus; those of Dr. Bradley (1723, 1747) describing his discovery of aberration and nutation; besides the accounts of Sir William Herschel's astronomical observations. The principal optical papers are those which contain Newton's experiments on light (1672); Dr. Halley's paper (1693) containing formulæ for finding the foci of optical glasses; Mr. Dollond's account (1758) of his discovery of the achromatic telescope; and the papers on the recent discovery of the polarity of light. To dynamics belong the papers containing the mathematical theory of the collision of bodies, by Dr. Wallis, Sir C. Wren, and Mr. Huygens (1668-9); Mr. Smeaton's paper (1776) on the quantity of mechanical power necessary for giving different degrees of velocity to heavy bodies; with those of Mr. Landen (1777) and Mr. Vince (1780) on the rotatory motion of bodies. There are comparatively few papers on the subject of mechanics: among them are those of Huygens and Hooke (1675) on watches: a valuable paper (1785) by Mr. Vince on friction; and one by Mr. Hodgkinson (1840) on the strength of cast-iron pillars. With respect to hydrodynamics and hydraulics, the most important are those of Mr. Canton (1762-4) on the compressibility and elasticity of fluids; and two by Mr. Atwood (1796-8) on the stability of ships. Mr. Smeaton has a valuable paper (1759) on the power of wind and water to turn machines; and Mr. Vince (1798) one on the resistance to bodies moving in fluids. There are several papers on the suspension of water in capillary tubes; and there is one by Mr. Beighton (1731), giving an account of the water-works at London. The papers on pneumatics consist chiefly of those which were contributed by Dr. Halley (1686), Sir George Shuckburgh, and General Roy (1777), on the measurement of heights by the barometer; with those of Greaves, Halley, Robins, Hutton, and Count Rumford, on the force of fired gunpowder. The principal papers relating to acoustics are, one by Dr. Denham (1768) on the velocity of sound; and one by Dr. Young (1800) on sound and light. Those which come under the head of navigation consist almost wholly of descriptions relating to the phenomena of tides. Almost every discovery in the science of electricity has The Transactions are rich in papers on Chemistry, which however are of a miscellaneous nature: among the more important are, the paper of Mr. Brand (1736) on hydrogen gas; that which states the experiments of Mr. Cavendish (1783) to determine the components of atmospherical air. There is a paper by Dr. Henry of Manchester (1797) on the expansion of carburetted hydrogen gas by electricity; and one by Mr. Kirwan (1786), containing an account of his experiments on sulphuretted hydrogen gas. There are many papers on the subject of Meteorology, but most of them are merely diaries of the weather: among them however is a paper by Dr. Heberden (1765) on the rate at which temperatures diminish in the atmosphere as the distances from the surface of the earth in crease. Among the papers which Dr. Thomson ranks as miscellaneous are the few which relate to Antiquities: these contain accounts of the ruins of Palmyra (1695), Pompey's pillar (1767), the catacombs of Rome and Naples (1760); and there are several papers on the discoveries made in Herculaneum. TRANSCENDENTAL, a mathematical term of description, the meaning of which is not very uniform. When any particular formula is incapable of being expressed by any particular range of algebraical symbols, it is, with respect to those symbols, transcendental, that is, it transcends or climbs beyond the power of these symbols. The word was perhaps first used by Leibnitz (Leipzig Acts,' 1686), who says, placet hoc loco, ut magis profutura dicamus, fontem aperire transcendentium quantitatum, cur nimirum quaedam problemata neque sint plana, neque solida, neque sursolida, aut ullius certigradus, sed omnem aequationem algebraicam transcendant.' Here then is the first meaning of the word; a transcendental problem is one the equation of which is infinitely high, or contains an infinite series of powers of an unknown quantity, so that its highest degree transcends every degree. To form an idea of what is now most commonly meant by transcendental, it will be desirable to recapitulate the steps by which algebra has arrived at its present state of expression; or rather, mathematical analysis, as those would say who do not like to call the differential calculus by the name of algebra. And first we have the state which preceded the time of Vieta, in which formulæ were mostly described in words, and the adoption of arbitrary symbols of quantity was only of casual occurrence. Next, we have the introduction of arbitrary symbols of quantity by Vieta, but not to the extent of using arbitrary numbers of multiplications, or algebraical exponents. Here what we now call an was transcendental; Vieta could have described as by a cubo-cubum, or a by a quadrato-quadrato-cubum, but a" had neither name nor symbol. Thirdly, we have the stage which began with Harriot and Descartes, and which brought ordinary algebra into substantially its present form. During these periods however geometry and arithmetic, without help from algebra, 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. means between two given numbers, did not place those lines or means among the objects of algebra, and gave no clue to any algebraical properties. Fourthly, we have the short but interesting period in which, before the formal invention of fluxions or the differential calculus, infinite series began to be employed, and the transcendentals last alluded to ceased to be absolutely incapable of expression. This was the state in which Leibnitz found the science when he first proposed to distinguish between algebraical and transcendental problems. 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. Let be the inverse function of x, so that (4x) is x; then 4(x-1) is a solution of the ori ginal equation, or px = 4 (4-1 x − 1) gives px.p’x = (x). Now 1 x is, when more than one root is used, inexpressible except by infinite series: that is, not merely inexpressible in common algebraical terms, but even with the assistance of logarithms and trigonometrical functions. Nevertheless, as particular cases of this solution, both ar and (b2x2) are found. 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 Sixthly, we have an alteration which it might have been gamma-function, sometimes the factorial function. Its supposed should have come long before, namely, the ex- expression is f ε x dx 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, &c. Strange as it may appear, this was never formally Nevertheless, as tables are now formed of its values, and as But when n is a fraction it can only be calculated by series. and steadily done till the time of Euler. And it is only in 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 logaangle 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 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 pa. 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:¥x = 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. 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 or 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 x=4 is satisfied by x = 2, or x=-2, because 2 x 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 4 into (x-2) (x + 2) = 0, with which it is identical, we then see that this product can only vanish when x - 2 or x + 2 vanishes; that is, when a is + 2 or 2. Transformations frequently leave a point unsettled which can only be determined by a subsequent species of experimental 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 frequently happens when the nature of the transformation is ascertained by means not of the expression to be transformed, but of one of its particular properties. For in 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+qx + rx2 +. and by the use of the property above alluded to, it is found that the series must be of the form 1 + Ax + ‡ A2x2 + .... But, as noticed in SYNTHESIS, all that is here proved is, that if a be capable of expansion in integer powers of a, 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. μ 1. Both systems oblique. Let x and y be the old coordinates of a point, x' and y' the new ones. Let 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 a with a means the amount of revolution which would bring the positive part of r into the direction of the positive part of ', the revolution being made in the positive direction. sin (0-0) '+ sin (0-4) y' sin x-μ= P. C., No. 1567. sin 4. Both systems rectangular. Here and - are both right angles. x-μ=cos p.x'—sin p.y' y—v=sin p.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 coincide, we have only to make μ=0, v=0, and use the formulæ accordingly. 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 a1, 1, the new ones. Let λ, μ, be the old coordinates of the new origin, and let the angle made by x, and y, be ¿, that of y and z1 be , and that of 21 and i be n, which we may thus denote: 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: desire to learn the origin of modern practical astronomy. The object of the present article is to give such an a=B'y"-y'ẞ", B=y'a"-a'y", ya'ẞ"-B'a" account of the transit as will enable any one to use it a'=3"y-"B, B'=y"a-a"y, y=a"ß-B"a with tolerable success. Those who wish for more perfect a" By' -yẞ', B" ya' -ay', y" aß' -ẞa'. information must consult the introductions to the Greenwich, Königsburg, Dorpat, Cambridge, Edinburgh, &c. ObFor the mode in which these nine quantities are made servations. Our type will be the portable transit-instruto depend upon three, we must refer to works on me- ment, leaving the reader to accommodate what is here said chanics, in which such reduction is particularly useful. to the powers of his own instrument, or to the practice We avoid giving it here, because trifling differences exist of the observatory which he adopts for a model. in the manner of taking the quantities to functions of There are three principal parts expressed in the plate. which all the rest are to be reduced, so that no set of equa- The iron stand, carrying the Y's with their adjustments; tions can be given which can be called universal. As far the telescope, inserted at right angles through an axis with as we have gone, the expressions of all writers are the a small vertical circle for finding or verifying stars; and same, though the letters used are not always alike. the cross level. The stand is made of cast-iron, and should 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 figured would be found inconveniently heavy if the instru another, and is sometimes beneficially employed for ment is often moved. The Y's are contained in brass reviving those who are nearly dying after severe hæmor- pieces, strongly united to the tops of the two uprights. rhage. The operation had long been used as a means of The left hand Y has a motion up and down, which is given experiment, and in the vain hope that by injecting the by a milled screw partially seen immediately under the blood of a healthy man or animal into the vessels of a pivot. The right hand Y is moved in azimuth by a screw, diseased one, the health of the latter would be restored; the milled head of which is seen projected upon the lantern. but it had rarely been employed for its only useful In portable instruments it is very convenient to have this purpose, till Dr. Blundell, after a long series of well-con- lateral or azimuthal adjustment made by screwing against ducted experiments on animals, proved that it might be a spring, as it is in this instrument. In fixed observatories safely and advantageously employed in men. His obser- the adjustment is made by two antagonist drawing screws, vations are published in his Physiological and Pathoone of which is tightened and the other loosened; and logical Researches ;' and since his revival of the ope- indeed this is the general construction of instruments of ration, the lives of many persons have been saved who every size, and is the most solid fixture. But it is so conwere, in all probability, dying from the loss of blood venient to be able to move the instrument at pleasure in during or after surgical operations, during gestation, and azimuth while actually looking through the telescope, that in other circumstances. The operation has indeed often we should strongly recommend the adoption of the counterfailed; it has often been unnecessarily performed; and its spring whenever the instrument is small, and is either to performance is not unaccompanied by danger to the be frequently shifted, or is not furnished with a meridian patient; but still there is sufficient evidence of its high mark. The spring must press pretty strongly against the utility in cases which, without it, would have been quite or screw, and there should be a clamping button in each nearly hopeless, to warrant its being resorted to under the adjustment, to keep all secure. guidance of a sound judgment. The chief instruments employed in the operation are a syringe, with double pipes, a basin of appropriate form, and a fine tube fixed on one of the pipes of the syringe. One of the veins of the arm of the patient being opened just sufficiently to admit the point of the tube, and fixed by a probe, blood must be drawn through a free opening in the vein of some healthy person, and as it flows into the basin must be slowly sucked up, without any mixture of air, by the syringe. When the syringe is filled and carefully cleared from all air by forcing blood up to the very point of the tube, the latter must be introduced into the patient's vein, and the blood steadily and slowly injected. Four or five ounces are often sufficient to revive a patient, and if they produce head-ache, flushings of the face, tendency to fainting, and other unpleasant symptoms, the transfusion should be arrested; but if not, the injection should be continued till it produces some good effect, or till a pint of blood has been transfused. Beyond this it is not safe to carry the operation, nor is it likely to be beneficial. A second or a third injection may be employed 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, The axis is made of two strong brass cones soldered on the central sphere. The sphere is cast hollow with two shoulders, over which the cones slip. As this is the most important part of the instrument, great care should be taken of the fitting before the axis is finally put together, and the symmetry of the parts as to the centre should be perfect. If the instrument is weak here, it is utterly worthless. In the older English instruments the centre was a cube, and that form is frequently adopted at present by continental artists. The transit at Bruxelles by Gambey, one of the largest and finest instruments in the world, is so constructed. The essential requisite however is symmetrical strength, and any shape is good which fulfils this condition. The pivots are soldered into the extremities of the cones, and are turned after the whole is fixed. One of them is pierced to admit light into the face of steel, which is less affected by wear. axis. In large instruments the pivots have an outer surGreater care is required to guard steel pivots from rust,* and the turning must be performed with a diamond cutter, as the hard knots to which steel is subject resist and jar the ordinary 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. When the surface is perfectly formed, the grinding should be discontinued, as a small difference of size in the 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 an annular plate in the centre, which makes an angle of 45° with the axis and with the telescope, and thus light enough is thrown down to the eye-end to illuminate the field very vividly, while the opening allows the rays from the object-end to pass without impediment. The quantity of light may be regulated by a contrivance for diminishing the aperture of the lantern, or by a shade passing between the lantern and the pivot. In some transits there is a contrivance for altering the angle of the central reflector in the body of the instrument; but this, although very handy, is objectionable, as affecting the symmetry of the instrument. In a thirty-inch transit the lantern is within reach, and may be twisted a little, so as to reduce the light at pleasure. The setting circle, with its level and clamp, are towards the illuminated end of the axis. The tail-piece, which is attached to the verniers and level, is held between the rounded ends of the two screws at a. By screwing one and loosening the other, the bubble of the level is brought to the middle, when the vernier points out the reading of the circle. There is a lens and reflector, for lighting and reading off the circle. The instrument here figured has a vernier which reads single minutes; but the vernier is inconveniently long for a fixed lens, and we should prefer reading to every 2', which is more than sufficiently near for finding or identifying stars. If the small circle is carefully |