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scribe, in very general terms, the manner moving body describe an ellipse round in which Newton, who was the first who its focus, which Kepler's observations systematically extended the laws of mo- had established to be the form of the ortion to the heavenly bodies, identified bits of the planets round the sun. The their results with the two remaining result of the inquiry shewed that this laws of Kepler. His “ Principles of curve required the same law of the force, Natural Philosophy" contain general varying inversely as the square of the propositions with regard to any law of distance, which therefore of course recentripetal force, but that which he sup- ceived additional confirmation. His meposed to be the true one in our system, is thod of doing this may, perhaps, be unexpressed in mathematical language, by derstood by referring to the last figure saying that the centripetal force varies but one, in which Cd, for instance, inversely as the square of the distance, representing the space fallen from which means, that if the force at any any point c towards s, in a given distance be taken for the unit of force, time, and the area CSD being proat half that distance, it is two times portional to the corresponding time, twice, or four times as strong; at one- the space through which the body would third the distance, three times thrice, or have fallen at C in any other time (which nine times as strong, and so for other would be greater, by Galileo's law, in distances. He shewed the probability proportion to the squares of the times), of this law in the first instance by com- might be represented by a quantity vaparing the motion of the moon with that rying directly, as Cd, and inversely in the of heavy bodies at the surface of the duplicate proportion of the triangular
earth. Taking LP area CSD, that is to say, proportional to M. to represent part of
Cd the moon's orbit de
if Dk be drawn from D
(SCX D k) scribed in one minute, the line P M between perpendicular on SC. If this polygon the orbit and the represent an ellipse, so that CD repre
tangent at L would sents a small arc of the curve, of which shew the space through which the central S is the focus, it is found by the nature force at the earth (assuming the above
of that curve, that is the same at principles of motion to be correct) would
(D k) draw the moon. From the known dis- all points of the curve, so that the law of tance and motion of the moon, this line variation of the force in the same ellipse PM is found to be about sixteen feet.
1 The distance of the moon is about sixty is represented solely by
If cd, times the radius of the earth, and therefore if the law of the central force in this &c. are drawn so that
is not the instance were such as has been supposed,
(D k) the force at the earth's surface would same at every point, the curve ceases to be 60 times 60, or 3600 times stronger, be an ellipse whose focus is at S, as and at the earth's surface, the central Newton has shewn in the same work. force would make a body fall through
(D k) : 3600 times 16 feet in one minute. Ga. The line to which is found to be
Cd lileo had already taught that the spaces through which a body would be made equal, is one drawn through the focus at to fall, by the constant action of the right angles to the longest axis of the same unvarying force, would be pro- ellipse till it meets the curve ;-this line portional to the squares of the times du- is called the latus rectum, and is a ring which the force was exerted, and third proportional to the two principal therefore according to these laws, a body at the earth's surface ought (since
Kepler's third law follows as an imthere are sixty seconds in a minute) to
mediate consequence of this determinafall through 16 feet in one second, which tion; for, according to what has been was precisely the space previously esta- already shown, the time of revolution blished by numerous experiments.
round the whole ellipse, or, as it is comWith this confirmation of the suppo
there is a point to which the name of centre is sition, Newton proceeded to the purely given, on uccount of peculiar properties belonging geometrical calculation of the law of to it : but the term " centripetal force' always re
fers to the place towards which the force is dicentripetal * force necessary to make a
rected, whether or not situated in the centre of the In many curves, as in the circle and ellipse, curve.
monly called, the periodic time, bears the loss will fall upon me. It will amount same ratio to the unit of time as the to giving me to understand, that I must whole area of the ellipse does to the area cease to profess Astronomy, after I have described in that unit. The area of the grown old in the belief of these opinions, whole ellipse is proportional in different having been hitherto gainsayed by no ellipses to the rectangle contained by the one,-and, in short, I must give up Austwo principal axes, and the area de- tria itself, if room is no longer to be left scribed in an unit of time is proportional in it for philosophical liberty." He was, to SC x Dk, that is to say, is in the sub- however, tranquillized, in a great degree, duplicate ratio of S C? Dk’, or a that “the book is only prohibited as
Dk by the reply of his friend, who told him when the force varies inversely as the contrary to the decree pronounced by the square of the distance SC; and in the holy office two years ago. This has been ellipse, as we have said already, this is partly occasioned by a Neapolitan monk equal to a third proportional to the (Foscarini), who was spreading these principal axes; consequently the pe- notions by publishing them in Italian, riodic times in different ellipses, which whence were arising dangerous conseare proportional to the whole areas of quences and opinions: and besides, Gathe ellipses directly, and the areas de- lileo was at the same time pleading his scribed in the unit of time inversely, cause at Rome with too much violence. are in the compound ratio of the rec- Copernicus has been corrected in the tangle of the axes directly, and subdu- same manner for some lines, at least in plicatly as a third proportional to the the beginning of his first book. But by axes inversely; that is to say, the squares obtaining a permission, they may be of these times are proportional to the read (and, as I suppose, this “ Epitome" cubes of the longest axes, which is also) by the learned and skilful in this Kepler's law.
science, both at Rome and throughout
all Italy. There is therefore no ground CHAPTER VIII.
for your alarm, either in Italy or Austria; The Epitome prohibited at Rome-Lo- only keep yourself within bounds, and The Epitome prohibited at Rome-Lo- put a guard upon your own passions.".
garithmic Tables -- Trial of Catharine Kepler-Kepler invited to Eng- ferent works on comets, beyond men
We shall not dwell upon Kepler's difland-Rudolphine Tables-Death
tioning that they were divided, on the Conclusion.
plan of many of his other publications, KEPLER'S “Epitome,” almost immedi- into three parts, Astronomical, Physical, ately on its appearance, enjoyed the ho- and Astrological. He maintained that nour of being placed by the side of the comets move in straight lines, with a work of Copernicus, on the list of books varying degree of velocity. Later theoprohibited by the congregation of the ries have shewn that they obey the same Index at Rome. He was considerably laws of motion as the planets, differing alarmed on receiving this intelligence, from them only in the extreme excenanticipating that it might occasion diffi- tricity of their orbits. In the second culties in publishing his future writings. book, which contains the Physiology of His words to Remus, who had communi- Comets, there is a passing remark that cated the news to him, are as follows:- comets come out from the remotest “ I learn from your letter, for the first parts of ether, as whales and monsters time, that my book is prohibited at Rome from the depth of the sea; and the sugand Florence. I particularly beg of you, gestion is thrown out that perhaps to send me the exact words of the cen- comets are something of the nature of sure, and that you will inform me whe- silkworms, and are wasted and conther that censure would be a snare for sumed in spinning their own tails. the author, if he were caught in Italy, or Among his other laborious employwhether, if taken, he would be enjoined ments, Kepler yet found time to cala recantation. It is also of consequence culate tables of logarithms, he having for me to know whether there is any been one of the first in Germany to apprechance of the same censure being ex- ciate the full importance of the facilities tended into Austria. For if this be so, they afford to the numerical calculator. not only shall I never again find a printer In 1618 he wrote to his friend Schickthere, but also the copies which the hard: “There is a Scottish Baron (whose bookseller has left in Austria at my de- name has escaped my memory), who has sire will be endangered, and the ultimate made a famous contrivance, hy which all need of multiplication and division is me forthwith to conceive the germ of a supplied by mere addition and subtrac- legitimate demonstration, which during tion; and he does it without sines. But that same winter I attempted, without even he wants a table of tangents *, and reference to lines or motion, or flow, or the variety, frequency, and difficulty of any other which I may call sensible the additions and subtractions, in some quality." cases, is greater than the labour of mul- Now to answer the question ; what is tiplying and dividing."
the use of logarithms ? Exactly what ten Kepler dedicated his “Ephemeris” for years ago was announced by their author, 1620 to the author of this celebrated in- Napier, and which may be told in these vention, Baron Napier, of Merchistoun; words.—Wheresoever in common arithand in 1624, published what he called metic, and in the Rule of Three, cometwo “Chilias Logarithmorum,” containing numbers to be multiplied together, there the Napierian logarithms of the quotients the sum of the logarithms is to be taken ; of 100,000 divided by the first ten num- where one number is to be divided by bers, then proceeding by the quotients of another, the difference; and the numevery ten to 100, and' by hundreds to ber corresponding to this sum or differ100,000. In the supplement published the ence, as the case may be, will be the following year, is a curious notice of the required product or quotient. This, manner in which this subtle contrivance
say, is the use of logarithms. But was at first received: “In the year 1621, in the same work in which I gave when I had gone into Upper Austria, and the demonstration of the principles, I had conferred everywhere with those could not satisfy the unfledged arithskilled in mathematics, on the subject of metical chickens, greedy of facilities, Napier's logarithms, I found that those and gaping with their beaks wide whose prudence had increased, and open, at the mention of this use, as whose readiness had diminished, through if to bolt down every particular gobbet, age, were hesitating whether to adopt till they are crammed with my preceptithis new sort of numbers, instead of cles." a table of sines; because they said The year 1622 was marked by the cait was disgraceful to a professor of tastrophe of a singular adventure which mathematics to exult like a child at befell Kepler's mother, Catharine, then some compendious method of working, nearly seventy years old, and by which and meanwhile to admit a form of cal- he had been greatly harassed and anculation, resting on no legitimate proof, noyed during several years. From her and which at some time might entangle youth she had been noted for a rude and us in error, when we least feared it. passionate temper, which on the present They complained that Napier's demon- occasion involved her in serious diffistration rested on a fiction of geometri- culties. One of her female acquaintcal motion, too loose and slippery for a ance, whose manner of life had been by sound method of reasonable demonstra
no means unblemished, was attacked tion to be founded on itt. “This led after a miscarriage by violent head
aches, and Catharine, who had often * The meaning of this passage is not very clear:
taken occasion to sneer at her notoKepler evidently had seen and used logarithms at the time of writing this letter; yet there is nothing rious reputation, was accused with havin the method to justify this expression,-“ At ing produced these consequences, by
* This was the objection originally made to the administration of poisonous potions. Newton's " Fluxions," and in fact, Napier's idea of She repelled the charge with violence, ceiving quantities. This may be seen at once from and instituted an action of scandal against
this person, but was unlucky (according I Def. A line is said to increase uniformly, when to Kepler's statement) in the choice of a
the point by which it is described passes
through equal intervals, in equal times. young doctor, whom she employed as 2 Def. A line is said to diminish to a shorter one her advocate. Considering the suit to be
proportionally, when the point passing along
very instructive, he delayed its terminational to the remainder.
tion during five years, until the judge 6 Def. The logarithm of any sine is the number
before whom it was tried was displaced. most nearly denoting the line, which has increased uniformly, whilst the radius has He was succeeded by another, already indiminished to that sine proportionally, the disposed against Catharine Kepler, who initial velocity being the same in both motions. (Mirifici logarithmorum canonis
on some occasion had taunted him with descriptio, Edinburgi 1614.)
his sudden accession to wealth from a This last definition contains what lieshould now call the differential equation between a number
very inferior situation. Her opponent, and the logarithm of its reciprocal.
aware of this advantage, turned the ta
bles on her, and in her turn became the this invitation an affront was put upon accuser. The end of the matter was, Kepler by his early patrons, the States that in July, 1620, Catharine was im- of Styria, who ordered all the copies of prisoned, and condemned to the torture. his “Calendar," for 1624, to be publicly Kepler was then at Linz, but as soon burnt. Kepler declares that the reason as he learned his mother's danger, hur- of this was, that he had given preceried to the scene of trial. He found the dence in the title-page to the States of charges against her supported only by Upper Ens, in whose service he then evidence which never could have been was, above Styria. As this happened listened to, if her own intemperate con- during his absence in Wirtemberg, it was duct had not given advantage to her immediately coupled by rumour with adversaries. He arrived in time to save his hasty departure from Linz: it was her from the question, but she was not said that he had incurred the Emperor's finally acquitted and released from pri- displeasure, and that a large sum was son till November in the following year. set upon his head. At this period MatKepler then returned to Linz, leaving thias had been succeeded by Ferdibehind him his mother, whose spirit nand III., who still continued to Kepler seemed in no degree broken by the un- his barren title of imperial mathemaexpected turn in the course of her liti- tician. gation. She immediately commenced In 1624 Kepler went to Vienna, in a new action for costs and damages the hopes of getting money to complete against the same antagonist, but this the Rudolphine Tables, but was obliged to was stopped by her death, in April 1622, be satisfied with the sum of 6000 forins in her seventy-fifth year.
and with recommendatory letters to the In 1620 Kepier was visited by Sir States of Suabia, from whom he also Henry Wotton, the English ambassador collected some money due to the em• at Venice, who finding him, as indeed peror. On his return he revisited the he might have been found at every period University of Tubingen, where he found of his life, oppressed by pecuniary diffi- his old preceptor, Mästlin, still alive, culties, urged him to go over to England, but almost worn out with old age. where he assured him of a welcome Mästlin had well deserved the regard and 'honourable reception; but Kepler Kepler always appears to have entercould not resolve upon the proposed tained for him; he had treated him with journey, although in his letters he often great liberality whilst at the University, returned to the consideration of it. In where he refused to receive any remuneone of them, dated a year later, he says, ration for his instruction. Kepler took “The fires of civil war are raging in every opportunity of shewing his gratiGermany—they who are opposed to the tude; even whilst he was struggling with honour of the empire are getting the poverty he contrived to send his old upper hand-everything in my neigh, master a handsome silver cup, in acbourhood seems abandoned to flame and knowledging the receipt of which Mästdestruction. Shall I then cross the sea, lin says, " Your mother had taken it whither Wotton invites me? I, a Ger. into her head that you owed me two man? a lover of firm land ? who dread hundred florins, and had brought fifteen the confinement of an island ? who pre- florins and a chandelier towards reducing sage its dangers, and must drag along the debt, which I advised her to send to with me my little wife and flock of chil- you. I asked her to stay to dinner, which dren ? Besides my son Louis, now she refused : however, we handselled thirteen years old, I have a marriage- your cup, as you know she is of a thirsty able daughter, a two-year old son by my temperament." second marriage, an infant daughter, and The publication of the Rudolphine its mother but just recovering from Tables, which Kepler always had so her confinement. Six years later, he much at heart, was again delayed, notsays again,—“As soon as the Rudol- withstanding the recent grant, by the phine Tables are published, my desire will disturbances arising out of the two parbe to find a place where I can lecture ties into which the Reformation had on them to a considerable assembly; if divided the whole of Germany. Kepler's possible, in Germany; if not, why then library was sealed up by desire of the in Italy, France, the Netherlands, or Jesuits, and nothing but his connexion England, provided the salary is ade- with the Imperial Court secured to him quate for a traveller."
his own personal indemnity. Then fol. In the same year in which he received lowed a popular insurrection, and the peasantry blockaded Linz, so that it was for by this means it may come to be not until 1627 that these celebrated tables corrected." finally made their appearance, the ear- This is probably one of the earliest liest calculatel on the supposition that announcements of the method of deterthe planets move in elliptic orbits. mining longitudes by occultations; the Ptolemy's tables had been succeeded by imperfect theory of the moon long rethe “ Alphonsine," so called from Al- mained a principal obstacle to its intro. phonso. King of Castile, who, in the duction in practice. Another interesting thirteenth century, was an enlightened passage connected with the same object patron of astronomy. After the disco. may be introduced here. In a leiter to veries of Copernicus, these again made his friend Cruger, dated in 1616, Kepway for the Prussian, or Prutenic tables, ler says: “You propose a method of calculated by his pupils Reinhold and observing the distances of places by sunRheticus. These remained in use till dials and automata. It is good, but needs the observations of Tycho Brahe showed a very accurate practice, and confidence their insufficiency, and Kepler's new in those who have the care of the clocks. theories enabled him to improve upon Let there be only one clock, and let it them. The necessary types for these be transported; and in both places let tables were cast at Kepler's own expense. meridian lines be drawn with which the They are divided into four parts, the clock may be compared when brought. first and third containing a variety of The only doubt remaining is, whether a logarithmic and other tables, for the greater error is likely from the unequal purpose of facilitating astronomical cal- tension in the automaton, and from its culations. In the second are tables of motion, which varies with the state of the elements of the sun, moon, and the air, or from actually measuring the planets. The fourth gives the places of distances. For if we trust the latter, 1000 stars as determined by Tycho, and we can easily determine the longitudes by also at the end his table of refractions, observing the differences of the height which appears to have been different for of the pole." the sun, moon, and stars. Tycho Brahe In an Appendix to the Rudolphine assumed the horizontal refraction of the Tables, or, as Kepler calls it," an sun to be 7' 30', of the moon 8', and of alms doled out to the nativity casters," the other stars 3'. He considered all he has shown how they may use his refraction of the atmosphere to be in- tables for their astrological predictions. sensible above 45° of altitude, and Everything in his hands became an even at half that altitude in the case of allegory; and on this occasion he says, the fixed stars. A more detailed ac. -"Astronomy is the daughter of Ascount of these tables is here obviously trology, and this modern Astrology, unsuitable: it will be sufficient to say again; is the daughter of Astronomy, merely, that if Kepler had done nothing bearing something of the lineaments of in the course of his whole life but con- her grandmother; and, as I have alstruct these, he would have well earned ready said, this foolish daughter, Astrothe title of a most useful and indefati- logy, supports her wise but needy mother, gable calculator.
Astronomy, from the profits of a profesSome copies of these tables have pre- sion not generally considered creditfixed to them a very remarkable map, able." divided by hour lines, the object of Soon after the publication of these which is thus explained :
tables, the Grand Duke of Tuscany sent "The use of this nautical mapis, that him a golden chain; and if we rememif at a given hour the place of the moon ber the high credit in which Galileo is known by its edge being observed to stood at this time in Florence, it does touch any known star, or the edges of not seem too much to attribute this the sun, or the shadow of the earth; honourable mark of approbation to his and if that place shall (if necessary) be representation of the value of Kepler's reduced from apparent to real by clear- services to astronomy. This was soon ing it of parallax; and if the hour at followed by a new and final change in his Uraniburg be computed by the Rudol- fortunes. He received permission from phine tables, when the moon occupied the emperor to attach himself to the that true place, the difference will show celebrated Duke of Friedland, Albert the observer's meridian, whether the Wallenstein, one of the most remark. picture of the shores be accurate or not, able men in the history of that time,