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Miss H. Johnson has undertaken the laborious and tedious task of reading the whole book in manuscript as well as in proof, and of verifying the cross-references. Miss F. Hardcastle, of Girton College, has also read the proofs, and verified most of the numerical calculations, as well as the cross-references. To both I am indebted for the detection of a large number of obscurities in expression, as well as of clerical and other errors and of misprints. Miss Johnson has also saved me much time by making the Index of Names, and Miss Hardcastle has rendered me ix a further service of great value by drawing a considerable number of the diagrams.

I am also indebted to Mr. Inglis, of this College, for fig. Wesley, of the Royal Astronomical Society, for various references to the literature of the subject, and in particular for help in obtaining access to various illustrations. I am further indebted to the following bodies and individual astronomers for permission to reproduce photographs and drawings, and in some cases also for the gift of copies of the originals: the Council of the Royal Society, the Council of the Royal Astronomical Society, the Director of the Lick Observatory, the Director of the Instituto Geographico-Militare of Florence, Professor Barnard, Major Darwin, Dr.

Gill, M. Janssen, M. Loewy, Mr. Maunder, Mr. Pain, Professor E. Pickering, Dr. Schuster, Dr. Max Wolf. Astronomy is the science which treats of the sun, the moon, the stars, and other objects such as comets which are seen in the sky. It deals to some extent also with the earth, but only in so far as it has properties in common with the heavenly bodies.

In early times astronomy was concerned almost entirely with the observed motions of the heavenly bodies. At a later stage astronomers were able to discover the distances and sizes of many of the heavenly bodies, and to weigh some of them; and more recently they have acquired a considerable amount of knowledge as to their nature and the material of which they are made. We know nothing of the beginnings of astronomy, and can only conjecture how certain of the simpler facts of the science—particularly those with a direct influence on human life and comfort—gradually became familiar to early mankind, very much as they are familiar to modern savages.

With these facts it is convenient to begin, taking them in the order in which they most readily present themselves to any ordinary observer. The sun is daily seen to rise in the eastern part of the sky, to travel across the sky, to reach its highest position in the south in the middle of the day, then to sink, and finally to set in the western part of the sky.

On Mach's Theories

But its daily path across the sky is not always the same: the points of the horizon at which it rises and sets, its height in the sky at midday, and the time from sunrise to sunset, all go through a series of changes, which are accompanied by changes in the weather, in vegetation, etc. But while the sun always appears as a bright circular disc, the next most conspicuous of the heavenly bodies, the moon, undergoes changes of form which readily strike the observer, and are at once seen to take place in a regular order and at about the same intervals of time.

A little more care, however, is necessary in order to observe the connection between the form of the moon and her position in the sky with respect to the sun. Thus when the moon is first visible soon after sunset near the place where the sun has set, her form is a thin crescent cf. Next night the moon is farther from the sun, the crescent is thicker, and she sets later; and so on, until after rather less than a week from the first appearance of the crescent, she appears as a semicircular disc, with the flat side turned away from the sun.

The semicircle enlarges, and after another week has grown into a complete disc; the moon is now nearly in the opposite direction to the sun, and therefore rises about at sunset and sets about at sunrise. She then begins to approach the sun on the other side, rising before it and setting in the daytime; her size again diminishes, until after another week she is again semicircular, the flat side being still turned away from the sun, but being now turned towards the west instead of towards the east.

The semicircle then becomes a gradually diminishing crescent, and the time of rising 3 approaches the time of sunrise, until the moon becomes altogether invisible. After two or three nights the new moon reappears, and the whole series of changes is repeated. The different forms thus assumed by the moon are now known as her phases ; the time occupied by this series of changes, the month, would naturally suggest itself as a convenient measure of time; and the day, month, and year would thus form the basis of a rough system of time-measurement. From a few observations of the stars it could also clearly be seen that they too, like the sun and moon, changed their positions in the sky, those towards the east being seen to rise, and those towards the west to sink and finally set, while others moved across the sky from east to west, and those in a certain northern part of the sky, though also in motion, were never seen either to rise or set.

Although anything like a complete classification of the stars belongs to a more advanced stage of the subject, a few star groups could easily be recognised, and their position in the sky could be used as a rough means of measuring time at night, just as the position of the sun to indicate the time of day. To these rudimentary notions important additions were made when rather more careful and prolonged observations became possible, and some little thought was devoted to their interpretation. Several peoples who reached a high stage of civilisation at an early period claim to have made important progress in astronomy.

Greek traditions assign considerable astronomical knowledge to Egyptian priests who lived some thousands of years B. On the other hand, the earliest recorded astronomical observation the authenticity of which may be accepted without scruple belongs only to the 8th century B. For the purposes of this book it is not worth while to make any attempt to disentangle from the mass of doubtful tradition and conjectural interpretation of inscriptions, bearing on this early astronomy, the few facts which lie embedded therein; and we may proceed at once to give some account of the astronomical knowledge, other than that already dealt with, which is discovered in the possession of the earliest really historical astronomers—the Greeks—at the beginning of their scientific history, leaving it an open question what portions of it were derived from Egyptians, Chaldaeans, their own ancestors, or other sources.

If an observer looks at the stars on any clear night he sees an apparently innumerable 1 host of them, which seem to lie on a portion of a spherical surface, of which he is the centre. This spherical surface is commonly spoken of as the sky, and is known to astronomy as the celestial sphere. The visible part of this sphere is bounded by the earth, so that only half can be seen at once; but only the slightest effort of the imagination is required to think of the other half as lying below the earth, and containing other stars, as well as the sun.

This sphere appears to the observer to be very large, though he is incapable of forming any precise estimate of its size. Most of us at the present day have been taught in childhood that the stars are at different distances, and that this sphere has in consequence no real existence. The early peoples had no knowledge of this, and for them the celestial sphere really existed, and was often thought to be a solid sphere of crystal. Moreover modern astronomers, as well as ancient, find it convenient for very many purposes to make use of this sphere, though it has no material existence, as a means of representing the directions in which the heavenly bodies are seen and their motions.

For all that direct observation 5 can tell us about the position of such an object as a star is its direction ; its distance can only be ascertained by indirect methods, if at all. When we speak, therefore, of a star as being at a point s on the celestial sphere, all that we mean is that it is in the same direction as the point s , or, in other words, that it is situated somewhere on the straight line through O and S. The advantages of this method of representing the position of a star become evident when we wish to compare the positions of several stars.

The difference of direction of two stars is the angle between the lines drawn from the eye to the stars; e. But if we represent the stars by the corresponding points p , q , r , s on the celestial sphere, then by an obvious property of the sphere the angle P O Q which is the same as p O q is less or greater than the angle R O S or r O s according as the arc joining p q on the sphere is less or greater than the arc joining r s , and in the same proportion; if, for example, the angle R O S is twice as great as the angle P O Q , so also is the arc p q twice as great as the arc r s.

We may therefore, in all questions relating only to the directions of the stars, replace the angle between the directions of two stars by the arc joining the corresponding points on the celestial sphere, or, in other words, by the distance between these points on the celestial sphere. But such arcs on a sphere are easier both to estimate by eye and to treat geometrically than angles, and the use of the celestial sphere is therefore of great value, apart from its historical origin. It is important to note that this apparent distance of two stars, i. In the figure, for example, Q is actually much nearer to S than it is to P , but the apparent distance measured by the arc q s is several times greater than q p.

The apparent distance of two points on the celestial sphere is measured numerically by the angle between the lines joining the eye to the two points, expressed in degrees , minutes , and seconds. We might of course agree to regard the celestial sphere as of a particular size, and then express the distance between two points on it in miles, feet, or inches; but it is practically very inconvenient to do so.

To say, as some people occasionally do, that the distance between two stars is so many feet is meaningless, unless the supposed size of the celestial sphere is given at the same time. It has already been pointed out that the observer is always at the centre of the celestial sphere; this remains 7 true even if he moves to another place. A sphere has, however, only one centre, and therefore if the sphere remains fixed the observer cannot move about and yet always remain at the centre.

The old astronomers met this difficulty by supposing that the celestial sphere was so large that any possible motion of the observer would be insignificant in comparison with the radius of the sphere and could be neglected. It is often more convenient—when we are using the sphere as a mere geometrical device for representing the position of the stars—to regard the sphere as moving with the observer, so that he always remains at the centre.

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Moreover a little careful observation would have shown that the motions of the stars in different parts of the sky, though at first sight very different, were just such as would have been produced by the celestial sphere—with the stars attached to it—turning about an axis passing through the centre and through a point in the northern sky close to the familiar pole-star. This point is called the pole. As, however, a straight line drawn through the centre of a sphere meets it in two points, the axis of the celestial sphere meets it again in a second point, opposite the first, lying in a part of the celestial sphere which is permanently below the horizon.

This second point is also called a pole; and if the two poles have to be distinguished, the one mentioned first is called the north pole , and the other the south pole. The moon, like the stars, shares this motion of the celestial sphere and so also does the sun, though this 8 is more difficult to recognise owing to the fact that the sun and stars are not seen together. As other motions of the celestial bodies have to be dealt with, the general motion just described may be conveniently referred to as the daily motion or daily rotation of the celestial sphere.

A further study of the daily motion would lead to the recognition of certain important circles of the celestial sphere. Each star describes in its daily motion a circle, the size of which depends on its distance from the poles. The pole-star describes so small a circle that its motion can only with difficulty be detected with the naked eye, stars a little farther off the pole describe larger circles, and so on, until we come to stars half-way between the two poles, which describe the largest circle which can be drawn on the celestial sphere.

The circle on which these stars lie and which is described by any one of them daily is called the equator. By looking at a diagram such as fig. If a star, such as S , lies on the north side of the equator, i. Such a star is called a northern circumpolar star. A slight familiarity with the stars is enough to shew any one that the same stars are not always visible at the same time of night.

Rather more careful observation, carried out for a considerable time, is necessary in order to see that the aspect of the sky changes in a regular way from night to night, and that after the lapse of a year the same stars become again visible at the same time. The explanation of these changes as due to the motion of the sun on the celestial sphere is more difficult, and the unknown discoverer of this fact certainly made one of the most important steps in early astronomy.

If an observer notices soon after sunset a star somewhere in the west, and looks for it again a few evenings later at about the same time, he finds it lower down and nearer to the sun; a few evenings later still it is invisible, while its place has now been taken by some other star which was at first farther east in the sky. This star can in turn be observed to approach the sun evening by evening. Or if the stars visible after sunset low down in the east are 10 noticed a few days later, they are found to be higher up in the sky, and their place is taken by other stars at first too low down to be seen.

Such observations of stars rising or setting about sunrise or sunset shewed to early observers that the stars were gradually changing their position with respect to the sun, or that the sun was changing its position with respect to the stars. The changes just described, coupled with the fact that the stars do not change their positions with respect to one another, shew that the stars as a whole perform their daily revolution rather more rapidly than the sun, and at such a rate that they gain on it one complete revolution in the course of the year.

This can be expressed otherwise in the form that the stars are all moving westward on the celestial sphere, relatively to the sun, so that stars on the east are continually approaching and those on the west continually receding from the sun. But, again, the same facts can be expressed with equal accuracy and greater simplicity if we regard the stars as fixed on the celestial sphere, and the sun as moving on it from west to east among them that is, in the direction opposite to that of the daily motion , and at such a rate as to complete a circuit of the celestial sphere and to return to the same position after a year.

This annual motion of the sun is, however, readily seen not to be merely a motion from west to east, for if so the sun would always rise and set at the same points of the horizon, as a star does, and its midday height in the sky and the time from sunrise to sunset would always be the same. We have already seen that if a star lies on the equator half of its daily path is above the horizon, if the star is north of the equator more than half, and if south of the equator less than half; and what is true of a star is true for the same reason of any body sharing the daily motion of the celestial sphere.

George Biddell Airy

This great circle is now called the ecliptic because eclipses take place only when the moon is in or near it , and the angle at which it cuts the equator is called the obliquity of the ecliptic. The Chinese claim to have measured the obliquity in B. The truth of this statement may reasonably be doubted, but on the other hand the statement of some late Greek writers that either Pythagoras or Anaximander 6th century B.

It must have been known with reasonable accuracy to both Chaldaeans and Egyptians long before. When the sun crosses the equator the day is equal to the night, and the times when this occurs are consequently known as the equinoxes , the vernal equinox occurring when the sun crosses the equator from south to north about March 21st , and the autumnal equinox when it crosses back about September 23rd. The points on the celestial sphere where the sun crosses the equator A , C in fig.

After the vernal equinox the sun in its path along the 12 ecliptic recedes from the equator towards the north, until it reaches, about three months afterwards, its greatest distance from the equator, and then approaches the equator again. The time when the sun is at its greatest distance from the equator on the north side is called the summer solstice , because then the northward motion of the sun is arrested and it temporarily appears to stand still.

Similarly the sun is at its greatest distance from the equator towards the south at the winter solstice. The points on the ecliptic B , D in fig. The earliest observers probably noticed particular groups of stars remarkable for their form or for the presence of bright stars among them, and occupied their fancy by tracing resemblances between them and familiar objects, etc.

We have thus at a very early period a rough attempt at dividing the stars into groups called constellations and at naming the latter. In some cases the stars regarded as belonging to a constellation form a well-marked group on the sky, sufficiently separated from other stars to be conveniently classed together, although the resemblance which the group bears to the object after which it is named is often very slight.

The seven bright stars of the Great Bear, for example, form a group which any observer would very soon notice and naturally make into a constellation, but the resemblance to a bear of these and the fainter stars of the constellation is sufficiently remote see fig. In very many cases the choice of stars seems to have been made in such an arbitrary manner, as to suggest that some fanciful figure was first imagined and that stars were then selected so as to represent it in some rough sort of way. Innumerable snakes twine through long and contorted areas of the heavens where no memory can follow them; bears, lions, and fishes, large and small, confuse all nomenclature.

The constellations as we now have them are, with the exception of a certain number chiefly in the southern skies which have been added in modern times, substantially those which existed in early Greek astronomy; and such information as we possess of the Chaldaean and Egyptian constellations shews resemblances indicating that the Greeks borrowed some of them. The names, as far as they are not those of animals or common objects Bear, Serpent, Lyre, etc.

Among the constellations which first received names were those through which the sun passes in its annual circuit of the celestial sphere, that is those through which the ecliptic passes. Thus 14 arose twelve zodiacal constellations , the names of which have come down to us with unimportant changes from early Greek times. In some cases individual stars also received special names, or were called after the part of the constellation in which they were situated, e. Sirius, the Eye of the Bull, the Heart of the Lion, etc. We have seen that the stars, as a whole, retain invariable positions on the celestial sphere, 6 whereas the sun and moon change their positions.

It was, however, discovered in prehistoric times that five bodies, at first sight barely distinguishable from the other stars, also changed their places. These five—Mercury, Venus, Mars, Jupiter, and Saturn—with the sun and moon, were called planets , 7 or wanderers, as distinguished from the fixed stars. Venus is conspicuous as the Evening Star or as the Morning Star. The discovery of the identity of the Evening and Morning Stars is attributed to Pythagoras 6th century B. Jupiter is at times as conspicuous as Venus at her brightest, while Mars and Saturn, when well situated, rank with the brightest of the fixed stars.

If we begin to watch a planet when it is moving eastwards among the stars, we find that after a time the motion becomes slower and slower, until the planet hardly seems to move at all, and then begins to move with gradually increasing speed in the opposite direction; after a time this westward motion becomes slower and then ceases, and the planet then begins to move eastwards again, at first slowly and then faster, until it returns to its original condition, and the changes are repeated.

When the planet is just reversing its motion it is said to be stationary , and its position then is called a stationary point. On the whole the planets advance from west to east and describe circuits round the celestial sphere in periods which are different for each planet. The explanation of these irregularities in the planetary motions was long one of the great difficulties of astronomy.

In this way the moon would be recognised as nearer than any of the other celestial bodies. No direct means being available for determining the distances, rapidity of motion was employed as a test of probable nearness. The stars being seen above us it was natural to think of the most distant celestial bodies as being the highest, and accordingly Saturn, Jupiter, and Mars being beyond the sun were called superior planets , as distinguished from the two inferior planets Venus and Mercury.

One of the purposes to which applications of astronomical knowledge was first applied was to the measurement of time. As the alternate appearance and disappearance of the sun, bringing with it light and heat, is the most obvious of astronomical facts, so the day is the simplest unit of time. According to this arrangement a day-hour was in summer longer than a 18 night-hour and in winter shorter, and the length of an hour varied during the year. At Babylon, for example, where this arrangement existed, the length of a day-hour was at midsummer about half as long again as in midwinter, and in London it would be about twice as long.

It was therefore a great improvement when the Greeks, in comparatively late times, divided the whole day into 24 equal hours. Other early nations divided the same period into 12 double hours, and others again into 60 hours. The next most obvious unit of time is the lunar month , or period during which the moon goes through her phases. A third independent unit is the year. Although the year is for ordinary life much more important than the month, yet as it is much longer and any one time of year is harder to recognise than a particular phase of the moon, the length of the year is more difficult to determine, and the earliest known systems of time-measurement were accordingly based on the month, not on the year.

They chose a year of days. The origin of the week is quite different from that of the month or year, and rests on certain astrological ideas about the planets. The first three are easily recognised in our Saturday, Sunday, and Monday; in the other days the names of the Roman gods have been replaced by their supposed Teutonic equivalents—Mercury by Wodan, Mars by Thues, Jupiter by Thor, Venus by Freia.

Eclipses of the sun and moon must from very early times have excited great interest, mingled with superstitious terror, and the hope of acquiring some knowledge of them was probably an important stimulus to early astronomical work. In fact even in the time of Anaxagoras 5th century B. One of the most remarkable of the Chaldaean contributions to astronomy was the discovery made at any rate several centuries B. It is probable that the discovery was made, not by calculations based on knowledge of the motions of the sun and moon, but by mere study of the dates on which eclipses were recorded to have taken place.

As, however, an eclipse of the sun unlike an eclipse of the moon is only visible over a small part of the surface of the earth, and eclipses of the sun occurring at intervals of eighteen years are not generally visible at the same place, it is not at all easy to see how the Chaldaeans could have established their cycle for this case, nor is it in fact clear that the saros was supposed to apply to solar as well as to lunar eclipses.

The saros may 20 be illustrated in modern times by the eclipses of the sun which took place on July 18th, , on July 29th, , and on August 9th, ; but the first was visible in Southern Europe, the second in North America, and the third in Northern Europe and Asia. To the Chaldaeans may be assigned also the doubtful honour of having been among the first to develop astrology , the false science which has professed to ascertain the influence of the stars on human affairs, to predict by celestial observations wars, famines, and pestilences, and to discover the fate of individuals from the positions of the stars at their birth.

A belief in some form of astrology has always prevailed in oriental countries; it flourished at times among the Greeks and the Romans; it formed an important part of the thought of the Middle Ages, and is not even quite extinct among ourselves at the present day. In the earlier period of Greek history one of the chief functions expected of astronomers was the proper regulation of the calendar. The Greeks, like earlier nations, began with a calendar based on the moon. In the time of Hesiod a year consisting of 12 months of 30 days was in common use; at a later date a year made up of 6 full months of 30 days and 6 empty months of 29 days was introduced.

To Solon is attributed the merit of having introduced at Athens, about B. This arrangement was further improved by the introduction, probably during the 5th century B. As, however, the Greeks laid some stress on beginning the month when the new moon was first visible, it was necessary to make from time to time arbitrary alterations in the calendar, and considerable confusion 22 resulted, of which Aristophanes makes the Moon complain in his play The Clouds , acted in B. A little later, the astronomer Meton born about B. The use of this cycle seems to have soon spread to other parts of Greece, and it is the basis of the present ecclesiastical rule for fixing Easter.

Owing to the same cause the early writers on agriculture e. Hesiod fixed the dates for agricultural operations, not by the calendar, but by the times of the rising and setting of constellations, i. The Roman calendar was in early times even more confused than the Greek. There appears to have been 23 at one time a year of either or days; tradition assigned to Numa the introduction of a cycle of four years, which brought the calendar into fair agreement with the sun, but made the average length of the month considerably too short.

Instead, however, of introducing further refinements the Romans cut the knot by entrusting to the ecclesiastical authorities the adjustment of the calendar from time to time, so as to make it agree with the sun and moon. According to one account, the first day of each month was proclaimed by a crier. A satisfactory reform of the calendar was finally effected by Julius Caesar during the short period of his supremacy at Rome, under the advice of an Alexandrine astronomer Sosigenes. The error in the calendar had mounted up to such an extent, that it was found necessary, in order to correct it, to interpolate three additional months in a single year 46 B.

The new system began with the year 45 B. To avoid returning to the subject, it may be convenient to deal here with the only later reform of any importance. The difference between the average length of the year as fixed by Julius Caesar and the true year is so small as only to amount to about one day in years. By the latter half of the 16th century the date of the vernal equinox was therefore about ten days earlier than it was at the time of the Council of Nice A.

The Gregorian Calendar , or New Style , as it was commonly called, was not adopted in England till , when 11 days had to be omitted; and has not yet been adopted in Russia and Greece, the dates there being now 12 days behind those of Western Europe. While their oriental predecessors had confined themselves chiefly to astronomical observations, the earlier Greek philosophers appear to have made next to no observations of importance, and to have been far more interested in inquiring into causes of phenomena.

Thales , the founder of the Ionian school, was credited by later writers with the introduction of Egyptian astronomy into Greece, at about the end of the 7th century B. On the other hand, some real progress seems to have been made by Pythagoras 11 and his followers. Pythagoras taught that the earth, in common with the heavenly bodies, is a sphere, and that it rests without requiring support in the middle of the universe.

Whether he had any real evidence in support of these views is doubtful, but it is at any rate a reasonable conjecture that he knew the moon to be bright because the sun shines on it, and the phases to be caused by the greater or less amount of the illuminated half turned towards us; and the curved form of the boundary between the bright and dark portions of the moon was correctly interpreted by him as evidence that the moon was spherical, and not a flat disc, as it appears at first sight.

Analogy would then probably suggest that the earth also was spherical. However this may be, the belief in the spherical form of the earth never disappeared from 25 Greek thought, and was in later times an established part of Greek systems, whence it has been handed down, almost unchanged, to modern times. This belief is thus 2, years older than the belief in the rotation of the earth and its revolution round the sun chapter IV. In Pythagoras occurs also, perhaps for the first time, an idea which had an extremely important influence on ancient and mediaeval astronomy.

Not only were the stars supposed to be attached to a crystal sphere, which revolved daily on an axis through the earth, but each of the seven planets the sun and moon being included moved on a sphere of its own. The distances of these spheres from the earth were fixed in accordance with certain speculative notions of Pythagoras as to numbers and music; hence the spheres as they revolved produced harmonious sounds which specially gifted persons might at times hear: this is the origin of the idea of the music of the spheres which recurs continually in mediaeval speculation and is found occasionally in modern literature.

At a later stage these spheres of Pythagoras were developed into a scientific representation of the motions of the celestial bodies, which remained the basis of astronomy till the time of Kepler chapter VII. The Pythagorean Philolaus , who lived about a century later than his master, introduced for the first time the idea of the motion of the earth: he appears to have regarded the earth, as well as the sun, moon, and five planets, as revolving round some central fire, the earth rotating on its own axis as it revolved, apparently in order to ensure that the central fire should always remain invisible to the inhabitants of the known parts of the earth.

The suggestion of such an important idea as that of the motion of the earth, an idea so 26 repugnant to uninstructed common sense, although presented in such a crude form, without any of the evidence required to win general assent, was, however, undoubtedly a valuable contribution to astronomical thought. It is well worth notice that Coppernicus in the great book which is the foundation of modern astronomy chapter IV. Three other Pythagoreans, belonging to the end of the 6th century and to the 5th century B.

Almost the only scientific Greek astronomer who believed in the motion of the earth was Aristarchus of Samos, who lived in the first half of the 3rd century B. He held that the sun and fixed stars were motionless, the sun being in the centre of the sphere on which the latter lay, and that the earth not only rotated on its axis, but also described an orbit round the sun.

Seleucus of Seleucia, who belonged to the middle of the 2nd century B. Unfortunately we know nothing of the grounds of this belief in either case, and their views appear to have found little favour among their contemporaries or successors. Plato about B. He condemned any careful study of the actual celestial motions as degrading rather than elevating, and apparently regarded the subject as worthy of attention chiefly on account of its connection with geometry, and because the actual celestial motions suggested ideal motions of greater beauty and interest.

This view of astronomy he contrasts with the popular conception, according to which the subject was useful chiefly for giving to the agriculturist, the navigator, and others a knowledge of times and seasons. The Sun, Mercury, and Venus are said to perform their revolutions in the same time, while the other planets move more slowly, statements which shew that Plato was at any rate aware that the motions of Venus and Mercury are different from those of the other planets. He also states that the moon shines by reflected light received from the sun.

Plato is said to have suggested to his pupils as a worthy problem the explanation of the celestial motions by means of a combination of uniform circular or spherical motions. He may be regarded as representative of the transition from speculative 28 to scientific Greek astronomy.

As in the schemes of several of his predecessors, the fixed stars lie on a sphere which revolves daily about an axis through the earth; the motion of each of the other bodies is produced by a combination of other spheres, the centre of each sphere lying on the surface of the preceding one. For the sun and moon three spheres were in each case necessary: one to produce the daily motion, shared by all the celestial bodies; one to produce the annual or monthly motion in the opposite direction along the ecliptic; and a third, with its axis inclined to the axis of the preceding, to produce the smaller motion to and from the ecliptic.

For each of the five planets four spheres were necessary, the additional one serving to produce the variations in the speed of the motion and the reversal of the direction of motion along the ecliptic chapter I. Thus the celestial motions were to some extent explained by means of a system of 27 spheres, 1 for the stars, 6 for the sun and moon, 20 for the planets. There is no clear evidence that Eudoxus made any serious attempt to arrange either the size or the time of revolution of the spheres so as to produce any precise agreement with the observed motions of the celestial bodies, though he knew with considerable accuracy the time required by each planet to return to the same position with respect to the sun; in other words, his scheme represented the celestial motions qualitatively but not quantitatively.

On the other hand, there is no reason to suppose that Eudoxus regarded his spheres with the possible exception of the sphere of the fixed stars as material; his known devotion to mathematics renders it probable that in his eyes as in those of most of the 29 scientific Greek astronomers who succeeded him the spheres were mere geometrical figures, useful as a means of resolving highly complicated motions into simpler elements. Eudoxus was also the first Greek recorded to have had an observatory, which was at Cnidus, but we have few details as to the instruments used or as to the observations made.

He was also an accomplished mathematician, and skilled in various other branches of learning. We have a tolerably full account of the astronomical views of Aristotle B. At the same time he treated the spheres as material bodies, thus converting an ingenious and beautiful geometrical scheme into a confused mechanism. Aristotle, in common with other philosophers of his time, believed the heavens and the heavenly bodies to be spherical.

Thus the visible portion of the moon is bounded by two planes passing nearly through its centre, perpendicular respectively to the lines joining the centre of the moon to those of the sun and earth. In the accompanying diagram, which represents a section through the centres of the sun S , earth E , and moon M , A B C D representing on a much enlarged scale a section of the moon itself, the portion D A B which is turned away from the sun is dark, while the portion A D C , being turned away from the observer on the earth, is in any case invisible to him.

The part of the moon which appears bright is therefore that of which B C is a section, or the portion represented by F B G C in fig. The breadth of this bright surface clearly varies with the relative positions of sun, moon, and earth; so that in the course of a month, during which the moon assumes successively the positions relative to sun and earth represented by 1, 2, 3, 4, 5, 6, 7, 8 in fig. Aristotle then argues that as one heavenly body is spherical, the others must be so also, and supports this conclusion by another argument, equally inconclusive to us, that a spherical form is appropriate to bodies moving as the heavenly bodies appear to do.

His proofs that the earth is spherical are more interesting. After discussing and rejecting various other suggested forms, he points out that an eclipse of the moon is caused by the shadow of the earth cast by the sun, and 32 argues from the circular form of the boundary of the shadow as seen on the face of the moon during the progress of the eclipse, or in a partial eclipse, that the earth must be spherical; for otherwise it would cast a shadow of a different shape.

A second reason for the spherical form of the earth is that when we move north and south the stars change their positions with respect to the horizon, while some even disappear and fresh ones take their place. For example, if a star is visible to an observer at A fig. Aristotle quotes further, in confirmation of the roundness of the earth, that travellers from the far East and the far West practically India and Morocco alike reported the presence of elephants, whence it may be inferred that the two regions in question are not very far apart.

He also makes use of some rather obscure arguments of an a priori character. Aristotle argues against the possibility of the revolution of the earth round the sun, on the ground that this motion, if it existed, ought to produce a corresponding apparent motion of the stars. We have here the first appearance of one of the most serious of the many objections ever brought against the belief in the motion of the earth, an objection really only finally disposed of during the 33 present century by the discovery that such a motion of the stars can be seen in a few cases, though owing to the almost inconceivably great distance of the stars the motion is imperceptible except by extremely refined methods of observation cf.

The question of the distances of the several celestial bodies is also discussed, and Aristotle arrives at the conclusion that the planets are farther off than the sun and moon, supporting his view by his observation of an occultation of Mars by the moon i. It is, however, difficult to see why he placed the planets beyond the sun, as he must have known that the intense brilliancy of the sun renders planets invisible in its neighbourhood, and that no occultations of planets by the sun could really have been seen even if they had been reported to have taken place. In astronomy, as in other subjects, Aristotle appears to have collected and systematised the best knowledge of the time; but his original contributions are not only not comparable with his contributions to the mental and moral sciences, but are inferior in value to his work in other natural sciences, e.

Natural History. Unfortunately the Greek astronomy of his time, still in an undeveloped state, was as it were crystallised in his writings, and his great authority was invoked, centuries afterwards, by comparatively unintelligent or ignorant disciples in support of doctrines which were plausible enough in his time, but which subsequent research was shewing to be untenable.

The advice which he gives to his readers at the beginning of his exposition of the planetary motions, to compare his views with those which they arrived at themselves or met with 34 elsewhere, might with advantage have been noted and followed by many of the so-called Aristotelians of the Middle Ages and of the Renaissance.

After the time of Aristotle the centre of Greek scientific thought moved to Alexandria. Founded by Alexander the Great who was for a time a pupil of Aristotle in B. These kings, especially the second of them, surnamed Philadelphos, were patrons of learning; they founded the famous Museum, which contained a magnificent library as well as an observatory, and Alexandria soon became the home of a distinguished body of mathematicians and astronomers.

Among the earlier members of the Alexandrine school were Aristarchus of Samos, Aristyllus , and Timocharis , three nearly contemporary astronomers belonging to the first half of the 3rd century B. A treatise of his On the Magnitudes and Distances of the Sun and Moon is still extant: he there gives an extremely ingenious method for ascertaining the comparative distances of the sun and moon. If, in the figure, E , S , and M denote respectively the centres of the earth, sun, and moon, the moon evidently appears to an observer at E half full when the angle E M S is a right angle. If when this is the case the angular distance between the centres of the sun and moon, i.

In fact, it being known by a well-known result in elementary geometry that the angles at E and S are together equal to a right angle, the angle at S is obtained by subtracting the angle S E M from a right angle. Aristarchus further estimated the apparent sizes of the sun and moon to be about equal as is shewn, for example, at an eclipse of the sun, when the moon sometimes rather more than hides the surface of the sun and sometimes does not quite cover it , and inferred correctly that the real diameters of the sun and moon were in proportion to their distances.

He appears also to have believed the distance of the fixed stars to be immeasurably great as compared with that of the sun. Both his speculative opinions and his actual results mark therefore a decided advance in astronomy. Timocharis and Aristyllus were the first to ascertain and to record the positions of the chief stars, by means of numerical measurements of their distances from fixed 36 positions on the sky; they may thus be regarded as the authors of the first real star catalogue, earlier astronomers having only attempted to fix the position of the stars by more or less vague verbal descriptions.

They also made a number of valuable observations of the planets, the sun, etc. Among the important contributions of the Greeks to astronomy must be placed the development, chiefly from the mathematical point of view, of the consequences of the rotation of the celestial sphere and of some of the simpler motions of the celestial bodies, a development the individual steps of which it is difficult to trace.

We have, however, a series of minor treatises or textbooks, written for the most part during the Alexandrine period, dealing with this branch of the subject known generally as Spherics , or the Doctrine of the Sphere , of which the Phenomena of the famous geometer Euclid about B.

In addition to the points and circles of the sphere already mentioned chapter I. Another important great circle was the meridian , passing through the zenith and the poles. The well-known Milky Way had been noticed, and was regarded as forming another great circle. There are also traces of the two chief methods in common use at the present day of indicating the position of a star on the celestial sphere, namely, by reference either to the equator or to the ecliptic. One of the applications of Spherics was to the construction of sun-dials, which were supposed to have been originally introduced into Greece from Babylon, but which were much improved by the Greeks, and extensively used both in Greek and in mediaeval times.

The proper graduation of sun-dials placed in various positions, horizontal, vertical, and oblique, required considerable mathematical skill. Much attention was also given to the time of the rising and setting of the various constellations, and to similar questions. The discovery of the spherical form of the earth led to a scientific treatment of the differences between the seasons in different parts of the earth, and to a corresponding division of the earth into zones. We have already seen that the height of the pole above the horizon varies in 38 different places, and that it was recognised that, if a traveller were to go far enough north, he would find the pole to coincide with the zenith, whereas by going south he would reach a region not very far beyond the limits of actual Greek travel where the pole would be on the horizon and the equator consequently pass through the zenith; in regions still farther south the north pole would be permanently invisible, and the south pole would appear above the horizon.

Any celestial body, therefore, the distance of which from the equator towards the north declination is less than P K , will cross the meridian to the south of the zenith, whereas if its declination be greater than P K , it will cross to the north of the zenith. This was known actually to be the case not very far south of Alexandria. It was similarly recognised that on the other side of the equator there must be a region in which the sun ordinarily cast shadows towards the south, but occasionally towards the north.

These two regions are the torrid zones of modern geographers. Similarly in the same regions the sun is in winter so near the south pole that for a time it remains continuously below the horizon. Regions in which this occurs our Arctic regions were unknown to Greek travellers, but their existence was clearly indicated by the astronomers. To Eratosthenes B. The distance between Alexandria and Syene 40 being known to be 5, stadia , Eratosthenes thus arrived at , stadia as an estimate of the circumference of the earth, a number altered into , in order to give an exact number of stadia for each degree on the earth.

It is evident that the data employed were rough, though the principle of the method is perfectly sound; it is, however, difficult to estimate the correctness of the result on account of the uncertainty as to the value of the stadium used. If, as seems probable, it was the common Olympic stadium, the result is about 20 per cent. An immense advance in astronomy was made by Hipparchus , whom all competent critics have agreed to rank far above any other astronomer of the ancient world, and who must stand side by side with the greatest astronomers of all time. We have also scarcely any information about his life.

He was born either at Nicaea in Bithynia or in Rhodes, in which island he erected an observatory and did most of his work. There is no evidence that he belonged to the Alexandrine school, though he probably visited Alexandria and may have made some observations there. Ptolemy mentions observations made by him in B. The period of his greatest activity must therefore have been about the middle of the 2nd century B. Apart from individual astronomical discoveries, his chief services to astronomy may be put under four heads.

He invented or greatly developed a special branch of mathe 41 matics, 21 which enabled processes of numerical calculation to be applied to geometrical figures, whether in a plane or on a sphere. Before no attempt had been made to form general equations for the motion or equilibrium of an elastic solid. Particular problems had been solved by special hypotheses. The earliest investigations of this century, by Thomas Young "Young's modulus of elasticity" in England, J. Binet in France, and G. Plana in Italy, were chiefly occupied in extending and correcting the earlier labours.

Between and the broad outline of the modern theory of elasticity was established. The boy was put out to a nurse, and he used to tell that when his father a common soldier came to see him one day, the nurse had gone out and left him suspended by a thin cord to a nail in the wall in order to protect him from perishing under the teeth of the carnivorous and unclean animals that roamed on the floor. Poisson used to add that his gymnastic efforts when thus suspended caused him to swing back and forth, and thus to gain an early familiarity with the pendulum, the study of which occupied him much in his maturer life.

His father destined him for the medical profession, but so repugnant was this to him that he was permitted to enter the Polytechnic School at the age of seventeen. His talents excited the interest of Lagrange and Laplace. At eighteen he wrote a memoir on finite differences which was printed on the recommendation of Legendre. He soon became a lecturer at the school, and continued through life to hold various government scientific posts and professorships.

He wrote on the mathematical theory of heat, capillary action, probability of judgment, the mathematical theory of electricity and magnetism, physical astronomy, the attraction of ellipsoids, definite integrals, series, and the theory of elasticity. He was considered one of the leading analysts of his time. His work on elasticity is hardly excelled by that of Cauchy, and second only to that of Saint-Venant. There is hardly a problem in elasticity to which he has not contributed, while many of his inquiries were new.

The equilibrium and motion of a circular plate was first successfully treated by him. Instead of the definite integrals of earlier writers, he used preferably finite summations. Poisson's contour conditions for elastic plates were objected to by Gustav Kirchhoff of Berlin, who established new conditions. But Thomson and Tait in their Treatise on Natural Philosophy have explained the discrepancy between Poisson's and Kirchhoff's boundary conditions, and established a reconciliation between them.

Important contributions to the theory of elasticity were made by Cauchy. To him we owe the origin of the theory of stress, and the transition from the consideration of the force upon a molecule exerted by its neighbours to the consideration of the stress upon a small plane at a point.

He anticipated Green and Stokes in giving the equations of isotropic elasticity with two constants. Weber was also the first to experiment on elastic after-strain. Other important experiments were made by different scientists, which disclosed a wider range of phenomena, and demanded a more comprehensive theory. Set was investigated by Gerstner — and Eaton Hodgkinson, while the latter physicist in England and Vicat — in France experimented extensively on absolute strength. Vicat boldly attacked the mathematical theories of flexure because they failed to consider shear and the time-element.

As a result, a truer theory of flexure was soon propounded by Saint-Venant.

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Poncelet advanced the theories of resilience and cohesion. He was called to Russia with Clapeyron and others to superintend the construction of bridges and roads. On his return, in , he was elected professor of physics at the Polytechnic School. Subsequently he held various engineering posts and professorships in Paris. As engineer he took an active part in the construction of the first railroads in France. He deserves much credit for his derivation and transformation of the general elastic equations, and for his application of them to double refraction.

Rectangular and triangular membranes were shown by him to be connected with questions in the theory of numbers. Neumann, Clerk Maxwell. Stokes, Wertheim, R. Clausius, Jellett, threw new light upon the subject of "rari-constancy" and "multi-constancy," which has long divided elasticians into two opposing factions. The uni-constant isotropy of Navier and Poisson had been questioned by Cauchy, and was now severely criticised by Green and Stokes. The charge brought by practical engineers, like Vicat, against the theorists led Saint-Venant to place the theory in its true place as a guide to the practical man.

Numerous errors committed by his predecessors were removed. He corrected the theory of flexure by the consideration of slide, the theory of elastic rods of double curvature by the introduction of the third moment, and the theory of torsion by the discovery of the distortion of the primitively plane section. His results on torsion abound in beautiful graphic illustrations. In case of a rod, upon the side surfaces of which no forces act, he showed that the problems of flexure and torsion can be solved, if the end-forces are distributed over the end-surfaces by a definite law.

Clebsch [66] extended the research to very thin rods and to very thin plates. Though often advantageous, this notation is cumbrous, and has not been generally adopted. Karl Pearson , professor in University College, London, has recently examined mathematically the permissible limits of the application of the ordinary theory of flexure of a beam. The mathematical theory of elasticity is still in an unsettled condition. Not only are scientists still divided into two schools of "rari-constancy" and "multi-constancy," but difference of opinion exists on other vital questions.

Boussinesq of Paris, and others. Sir William Thomson applied the laws of elasticity of solids to the investigation of the earth's elasticity, which is an important element in the theory of ocean-tides.

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If the earth is a solid, then its elasticity co-operates with gravity in opposing deformation due to the attraction of the sun and moon. Laplace had shown how the earth would behave if it resisted deformation only by gravity. Sir William Thomson combined the two results, and compared them with the actual deformation. Thomson, and afterwards G. This conclusion has been confirmed recently by Simon Newcomb, from the study of the observed periodic changes in latitude. For an ideally rigid earth the period would be days, but if as rigid as steel, it would be , the observed period being days.

Ibbetson, and F. Neumann, edited by O. Riemann's opinion that a science of physics only exists since the invention of differential equations finds corroboration even in this brief and fragmentary outline of the progress of mathematical physics. The undulatory theory of light, first advanced by Huygens, owes much to the power of mathematics: by mathematical analysis its assumptions were worked out to their last consequences. Thomas Young [95] — was the first to explain the principle of interference, both of light and sound, and the first to bring forward the idea of transverse vibrations in light waves.

Young's explanations, not being verified by him by extensive numerical calculations, attracted little notice, and it was not until Augustin Fresnel — applied mathematical analysis to a much greater extent than Young had done, that the undulatory theory began to carry conviction. Some of Fresnel's mathematical assumptions were not satisfactory; hence Laplace, Poisson, and others belonging to the strictly mathematical school, at first disdained to consider the theory. By their opposition Fresnel was spurred to greater exertion. Arago was the first great convert made by Fresnel.

When polarisation and double refraction were explained by Young and Fresnel, then Laplace was at last won over. But this was found to be in accordance with fact. The theory was placed on a sounder dynamical basis by the writings of Cauchy, Biot, Green, C. In the wave-theory, as taught by Green and others, the luminiferous ether was an incompressible elastic solid, for the reason that fluids could not propagate transverse vibrations. But, according to Green, such an elastic solid would transmit a longitudinal disturbance with infinite velocity.

Stokes remarked, however, that the ether might act like a fluid in case of finite disturbances, and like an elastic solid in case of the infinitesimal disturbances in light propagation. Fresnel postulated the density of ether to be different in different media, but the elasticity the same, while C. Neumann and McCullagh assume the density uniform and the elasticity different in all substances.

On the latter assumption the direction of vibration lies in the plane of polarisation, and not perpendicular to it, as in the theory of Fresnel. While the above writers endeavoured to explain all optical properties of a medium on the supposition that they arise entirely from difference in rigidity or density of the ether in the medium, there is another school advancing theories in which the mutual action between the molecules of the body and the ether is considered the main cause of refraction and dispersion.

Boussinesq, W. Sellmeyer, Helmholtz, E. Lommel, E. Ketteler, W. Neither this nor the first-named school succeeded in explaining all the phenomena. A third school was founded by Maxwell. He proposed the electro-magnetic theory, which has received extensive development recently. It will be mentioned again later. According to Maxwell's theory, the direction of vibration does not lie exclusively in the plane of polarisation, nor in a plane perpendicular to it, but something occurs in both planes—a magnetic vibration in one, and an electric in the other.

Fitzgerald and Trouton in Dublin verified this conclusion of Maxwell by experiments on electro-magnetic waves. Of recent mathematical and experimental contributions to optics, mention must be made of H. Rowland's theory of concave gratings, and of A. Michelson's work on interference, and his application of interference methods to astronomical measurements. In electricity the mathematical theory and the measurements of Henry Cavendish — , and in magnetism the measurements of Charles Augustin Coulomb — , became the foundations for a system of measurement.

The first complete method of measurement was the system of absolute measurements of terrestrial magnetism introduced by Gauss and Wilhelm Weber — and afterwards extended by Wilhelm Weber and F. Kohlrausch to electro-magnetism and electro-statics.

In the British Association and the Royal Society appointed a special commission with Sir William Thomson at the head, to consider the unit of electrical resistance. The commission recommended a unit in principle like W. Soon after, Laplace gave the celebrated differential equation,. The first to apply the potential function to other than gravitation problems was George Green — He introduced it into the mathematical theory of electricity and magnetism.

Green was a self-educated man who started out as a baker, and at his death was fellow of Caius College, Cambridge. In he published by subscription at Nottingham a paper entitled Essay on the application of mathematical analysis to the theory of electricity and magnetism. It escaped the notice even of English mathematicians until , when Sir William Thomson had it reprinted in Crelle's Journal , vols. It contained what is now known as "Green's theorem" for the treatment of potential.

The term potential function is due to Green.

Hamilton used the word force-function , while Gauss, who about secured the general adoption of the function, called it simply potential. Large contributions to electricity and magnetism have been made by William Thomson. He was born in at Belfast, Ireland, but is of Scotch descent. He and his brother James studied in Glasgow. From there he entered Cambridge, and was graduated as Second Wrangler in Thomson are a group of great men who were Second Wranglers at Cambridge.

At the age of twenty-two W. Thomson was elected professor of natural philosophy in the University of Glasgow, a position which he has held ever since. For his brilliant mathematical and physical achievements he was knighted, and in was made Lord Kelvin. His researches on the theory of potential are epoch-making.

What is called "Dirichlet's principle" was discovered by him in , somewhat earlier than by Dirichlet. We owe to Sir William Thomson new synthetical methods of great elegance, viz. By them he determined the distribution of electricity on a bowl, a problem previously considered insolvable. The distribution of static electricity on conductors had been studied before this mainly by Poisson and Plana.

In F. In W. Thomson predicted by mathematical analysis that the discharge of a Leyden jar through a linear conductor would in certain cases consist of a series of decaying oscillations. This was first established experimentally by Joseph Henry of Washington. William Thomson worked out the electro-static induction in submarine cables. The subject of the screening effect against induction, due to sheets of different metals, was worked out mathematically by Horace Lamb and also by Charles Niven. Weber's chief researches were on electrodynamics. Helmholtz in gave the mathematical theory of the course of induced currents in various cases.

Gustav Robert Kirchhoff [97] — investigated the distribution of a current over a flat conductor, and also the strength of current in each branch of a network of linear conductors. He was born near Edinburgh, entered the University of Edinburgh, and became a pupil of Kelland and Forbes. Routh being Senior Wrangler.

Maxwell then became lecturer at Cambridge, in professor at Aberdeen, and in professor at King's College, London. In he retired to private life until , when he became professor of physics at Cambridge. Maxwell not only translated into mathematical language the experimental results of Faraday, but established the electro-magnetic theory of light, since verified experimentally by Hertz. His first researches thereon were published in In appeared his great Treatise on Electricity and Magnetism.

He constructed the electro-magnetic theory from general equations, which are established upon purely dynamical principles, and which determine the state of the electric field. It is a mathematical discussion of the stresses and strains in a dielectric medium subjected to electro-magnetic forces.

The electro-magnetic theory has received developments from Lord Rayleigh, J. Thomson, H. Rowland, R. Glazebrook, H. Helmholtz, L. Boltzmann, O. Heaviside, J. Poynting, and others. Hermann von Helmholtz turned his attention to this part of the subject in He was born in at Potsdam, studied at the University of Berlin, and published in his pamphlet Ueber die Erhaltung der Kraft.

He became teacher of anatomy in the Academy of Art in Berlin. It was at Heidelberg that he produced his work on Tonempfindung. In he accepted the chair of physics at the University of Berlin. From this time on he has been engaged chiefly on inquiries in electricity and hydrodynamics. Weber, F. Neumann, Riemann, and Clausius, who had attempted to explain electrodynamic phenomena by the assumption of forces acting at a distance between two portions of the hypothetical electrical fluid,—the intensity being dependent not only on the distance, but also on the velocity and acceleration,—and the theory of Faraday and Maxwell, which discarded action at a distance and assumed stresses and strains in the dielectric.

His experiments favoured the British theory. He wrote on abnormal dispersion, and created analogies between electro-dynamics and hydrodynamics. Lord Rayleigh compared electro-magnetic problems with their mechanical analogues, gave a dynamical theory of diffraction, and applied Laplace's coefficients to the theory of radiation. Rowland made some emendations on Stokes' paper on diffraction and considered the propagation of an arbitrary electro-magnetic disturbance and spherical waves of light.

Electro-magnetic induction has been investigated mathematically by Oliver Heaviside, and he showed that in a cable it is an actual benefit. Heaviside and Poynting have reached remarkable mathematical results in their interpretation and development of Maxwell's theory. Most of Heaviside's papers have been published since ; they cover a wide field. One part of the theory of capillary attraction, left defective by Laplace, namely, the action of a solid upon a liquid, and the mutual action between two liquids, was made dynamically perfect by Gauss.

He stated the rule for angles of contact between liquids and solids. A similar rule for liquids was established by Ernst Franz Neumann. Chief among recent workers on the mathematical theory of capillarity are Lord Rayleigh and E. James Prescott Joule — determined experimentally the mechanical equivalent of heat. Helmholtz in applied the conceptions of the transformation and conservation of energy to the various branches of physics, and thereby linked together many well-known phenomena.

These labours led to the abandonment of the corpuscular theory of heat. The mathematical treatment of thermic problems was demanded by practical considerations. Thermodynamics grew out of the attempt to determine mathematically how much work can be gotten out of a steam engine. Sadi-Carnot , an adherent of the corpuscular theory, gave the first impulse to this. The principle known by his name was published in Though the importance of his work was emphasised by B. Clapeyron , it did not meet with general recognition until it was brought forward by William Thomson.

The latter pointed out the necessity of modifying Carnot's reasoning so as to bring it into accord with the new theory of heat. William Thomson showed in that Carnot's principle led to the conception of an absolute scale of temperature. In he published "an account of Carnot's theory of the motive power of heat, with numerical results deduced from Regnault's experiments. In the same month William John M. Rankine — , professor of engineering and mechanics at Glasgow, read before the Royal Society of Edinburgh a paper in which he declares the nature of heat to consist in the rotational motion of molecules, and arrives at some of the results reached previously by Clausius.

His proof of the second law is not free from objections. In March, , appeared a paper of William Thomson which contained a perfectly rigorous proof of the second law. He obtained it before he had seen the researches of Clausius. The statement of this law, as given by Clausius, has been much criticised, particularly by Rankine, Theodor Wand, P.

Tait, and Tolver Preston. Repeated efforts to deduce it from general mechanical principles have remained fruitless. The science of thermodynamics was developed with great success by Thomson, Clausius, and Rankine. As early as Thomson discovered the law of the dissipation of energy, deduced at a later period also by Clausius.

The latter designated the non-transformable energy by the name entropy , and then stated that the entropy of the universe tends toward a maximum. For entropy Rankine used the term thermodynamic function. Thermodynamic investigations have been carried on also by G. Hirn of Colmar, and Helmholtz monocyclic and polycyclic systems. Valuable graphic methods for the study of thermodynamic relations were devised in — by J.

Willard Gibbs of Yale College.