angles of the one are equal to the remaining angles of the other, each to each; viz, those to which the equal sides are opposite; that is, the angle ABC is equal to the angle DEF, and the angle ACB to the angle DFE. BADUY For, if the triangle ABC be applied to the triangle DEF, SO that the point в may be on the point E, and the straight line BA on the straight line ED, the point A shall coincide with the point D, because the straight line BA is equal to the straight line ED. Now, BA coinciding with ED, the straight line AC shall coincide with the straight line DE, because the angle BAC is equal, by hypothesis (i. e., by supposition), to the angle EDF. grecque hébraïque, used only of the Hebrew Also, the point c shall coincide with the point F, because, by tongue jouvencelle hypothesis, the straight line AC is equal to the straight line DF. But, by supraposition (placing upon), the point в coincides with the point E. Therefore, the base BC shall coincide with the base EF. For, the point B coinciding with the point E, and the point c with the point F, if the base в c does not coincide with the base E F, the two straight lines B C and E F, would enclose a space, which by I. Ax. 10, is impossible. Wherefore the base B c coincides with the base E F, and is, therefore, by I. Ax. 8, equal to it. Wherefore, also, the whole triangle ABC coincides with the whole triangle DEF, and is, therefore, equal to it. And the remaining angles of the one coinacide with the remaining angles of the other, and are therefore equal to them, each to each; viz., the angle ABC to the angle DEF, and the angle ACB to the angle DF E. Q. E, D. This proposition holds equally true, when the angles contained by the two sides, of the one triangle is the same as that contained by the two sides of the other; as, in the triangles FAC and GA B, see fig. 5; or, when the triangles have a common base, as in the triangles FBC and GCB, see fig. 5 next lesson; or when they have a common side. LESSONS IN GEOMETRY. In oala's Anilos al 3dr If two squares have one side of the one equal to one side of the other the squares are equal in all respects. In fig. (C) let ABCD and EFGH be two squares, which have one side AB A To produce the smaller of two given straight lines so that, with of the one equal to one side EF of the the part produced, it may be equal to the greater. EXERCISE ON PROP. III. BOOK I. other; these two squares are equal in B C F In fig. (B) let AB and c be the two given straight lines of which AB is the less. It is required to produce AB SO that, with the part produced, it may be equal to the greater c. G E From the point A in the straight line AB draw the straight line A D equal to the given straight line c (I. Prop. 2); and from the centre A with radius A D, describe the circle DEF (I. Post. 3), and produce A B to (I. Post. 2). Then the straight line A E is equal to c. For, by I. Def. 15, the straight line A E is equal to the straight line A D; and, by construction, the straight line c is equal to the straight line A D. Therefore, by I. Axiom 1, the straight line AE is equal to the straight line c, and the straight line AB has been produced to E, so that, with the part B E produced it is equal to c. Q. E. F.,-that is, Quod Erat Faciendum, Which Was To be done.* B Because A B is equal to EF, by hypothesis, and AB to Be by I. Def. 30, therefore, EF is equal to в c, by I. Ax. 1., but EF is equal to FG, by I. Def. 30; therefore, BC is equal F G, by I. Ax. 1. In the same manner, it may be shown that oD is equal des A1, Wherefore all the sides AB, BC, to GH, and HE to DA. CD, and DA of the square A B C D, are respectively equal to all the sides E F, FG, GH, and HE of the square EFGH. Also, by I. Ax. 11, all the angles A B C, B CD, CDA, and DA B, of the square A B C D, are respectively equal to all the angles EPG, FGH, GHE, and HEF of the square E F G H. Now, if the square ABCD be applied to the square EF GH so that the point a may be upon the point E, and the straight line A в upon the straight line EF, the point B will coincide with the point F, because B BO will coincide with the straight line FG, because the angle is equal to EF; and AB coinciding with EF, the straight line ABO has been proved equal to the angle EFG; also, the point a will coincide with the point o, because the straight line BC has been proved equal to the straight line FG. In the same manner it may be shown that the straight lines CD and DA will coincide with the straight lines GH and H E respectively, and the angles BCD, CDA, and D AB, with the angles FG H, GHA, and н E F, respectively. Therefore, the whole square A B CD coincides with the whole square E F G H in all respects, and the square ABCD is equal to the square EFGH, by I. Axiom 8. Q. E. D.* Harl In Dr. Thomson's edition of the "Elements," there is a corollary appended to this proposition, by which he is enabled to shorten the demonstration of the next, which from its difficulty to be understood by some minds, is the Pons Asinorum, the Asses Bridge. This corollary appears to us to need demonstration; this we shall give in our next lesson. It will then be seen how much it shortens and simplifies Euclid's demonstration of the Pons. This exercise, as well as the preceding one referred to in our last lesson, was performed correctly by J. H. Eastwood (Middleton); F. H. (Brightwell); E. C. Hughes (Islington); T. Bocock (Great Warley); A. Z. (St. Neot's); J. Bourton (Paddington); A. Coats (Glasgow); and others. This exercise was performed by J. H. Eastwood (Middleton); A. Z (St. Neots'); J. Bourton (Paddington); F. H. (Brightwell), T. Bocock (Great Warley); E. J. Bremner (Carlisle); A. Coats (Glasgow); E. C. Hughes (Islington); and others. ge RULE 1-When the given number is integral or whole, place unity or 1 under it, for the improper fraction required. The reason of this rule is evident; because if you divide any number by 1, the quotient is the same as the dividend, that is, the given number is not altered in value. Here, 12 and, 12 answer first; 84 12 X 7 the same as that of the given fraction, and then find the sum of this fraction and the given fraction, the result will be the fraction required. Or, which is the same thing, multiply the given whole number by the denominator of the given fraction, and to the product add its numerator, the sum placed over the said denominator will give the fraction required. The first part of this rule explains the reason of the second part, which is the rule usually given in books; see also definition 4, page 134. of of of of 7 9 8 8 14 5 9 15 48 5 of of 7 RULE 2.-When the given number is integral, and is required to have a given denominator. Make the given number an impro-rator into their component factors, and those which form the per fraction by rule 1, and then multiply both of its terms by the divisor or denominator into theirs, omit the factors which are comgiven denominator, the products will be the terms of the fraction mon to both, and use the remaining numbers instead; that is, required; see principle 1, p. 134, vol. II. multiplying the remaining numbers in the dividend or numerator for the new dividend or numerator, and the remaining numbers in the divisor or denominator for the new divisor or denominator; these products will produce the quotient or fraction required. Taking the three examples above given, they will be treated according to this rule, as follows: EXAMPLE. Reduce the number 12 to an improper fraction, by rule 1; and then, to an improper fraction whose denominator shall be 7, by rule 2. Given Compound Fractions. 1 X3 X4 X 7 15. An important principle much employed denominated concelling of factors, and of great utility in the practice of the preceding rule and many others in arithmetic, may be brought under the notice of the student at this point of his studies. If the product of one set of numbers is to be divided by the product of another set, or employed in a fractional form as above, the quotient in the one case or the fraction in the other, will not be altered, if we throw ext factors common to both sets of numbers; see principle 1, page 134. On this principle, we found the following rule for cancelling the factors in any such operations as the preceding. RULE.-Separate the numbers which form the dividend or nume Required Lowest terms. 3 X5 X 8 X 14 1680 20 51 3 of 44.P 9 = 7 1x3x4x7 EXERCISES. = 1. Reduce the whole numbers 25, 48, 301, 4,000, and 5,876,934, to improper fractions. 1x7 of 2. Reduce the preceding whole numbers to improper fractions whose denominators shall be 12, 6, 5, 4, and 2, respectively. of RULE 3.-When the given number is mixed, that is, composed of a whole number and a fraction. Reduce the given whole number, by rule 2, to an improper fraction whose denominator shall be In reference to the last example here, it is necessary to observe that 8 10 27 = = = Required Fractions. 1 is a factor of every number, and that it must be put down as such, when all the other factors are to be cancelled either above or below the line; or when both above and below, that is, when there is a perfect equality of factors. EXERCISES. Reduce the following compound fractions to simple fractions, both by the rule to problem V, and by cancelling factors: 1- of of of 4.- of of off of 2.- of of of 1% of 5.- of of of 9 3.- of of of 6.- of of 19 of 128. ADDITION OF FRACTIONS. PROBLEM VI. To add fractions, or mixed numbers together. RULE 1.-When the given fractions have a common deno. minator, find the sum of the numerators, and under it place the common denominator, for the sum required. The sum should be reduced to its lowest terms, and if it be an improper fraction, may be reduced to a whole or mixed number, by problem III. EXAMPLE. Find the sum of the fractions,,,, t, &,, and it . ON COM ET S.-No. II. 1828, 1832, &c. SINCE the appearance of the comet of Encké, in 1811, its course has been determined. The return of this comet was looked for in 1822, and this event was only realised at Paramatta, in New South Wales, where it was observed during the month of June. The epoch or instant of its passing the perihelion only differed by three hours from that predicted by M. Encké. Its return in September, 1825, was equally predicted and verified; it has since appeared in This comet, which is powerfully attracted by Mercury, presents much similarity to the planet Ceres in its elements, the inclination and the major axis or greatest diameter of Here, +++8+8+8+3=?==3}, the sum required. the orbit being the same as those of that planet; its periodic revoRULE 2.-If the given fractions have different denominators, lution is forty-six days less than that of Vesta; its perihelion reduce them by problem II. p. 267, to fractions of equal value falls within the orbit of Mercury; the aphelion falls between that having a common denominator, and apply the preceding rule to of Jupiter and the new planets. Is it possible that this comet is a these fractions, the sum will be the answer required. fragment of our supposed Pluto? (see No. 31, p. 62). As to the The reason of these rules is explained at p. 168, vol. II. comet of M. Biela, its course is also known; it was seen in 1806 EXAMPLE.-Fnd the sum of the following fractions: 3, 2, and 3. the mean duration of its revolution is six years and three quarters; and 1826, and was supposed to be the same that was seen in 1772; Here, by problem II. p. 267, we have 40 60 = 48 60 Whence, 133243, is the answer required. Or, 18+8+8+ 18-218, sum of the fractions. 12 118 324 11491% RULE 3.-When the given numbers are mixed, find the sum of the whole numbers by the rule of simple addition, p. 56, vol. I. and the sum of the fractions by the preceding rules; then add these sums together, and the result will be the whole sum required. Or, reduce the mixed numbers to improper fractions, and find their sum by the preceding rules. The reason of this rule is plain from the reasons assigned for the preceding rules. EXAMPLE. Find the sum of the following mixed numbers: 4, 7. 12, 118, 32, and 1149. Here, 4+7+12+1189+32+1149=1322 sum of the whole numbers. And by Problem II., p. 267, we have 1 = 60 99 125 || || || || || || 96 Numerators, 45 133 If it be not impossible that some comet should one day strike against the earth, there are at least a thousand probabilities that this event will not happen. It must be a very extraordinary occurrence in which two bodies so small, moving in the immensity of space, with every variety of velocity, in orbits of every kind of dimensions and with every degree of inclination, should happen to strike against each other. But this indefinitude of occurrence disappears before another, namely, that of time; indefinite duration admits of the conception of all possible events being realised; and the effect of such a concussion would, indeed, be terrible. Laplace has the following remarks on this subject:-"It is easy to represent the effects of the shock given to the earth by a comet. The 10 Whence, ++128+128+128+128=8=4,5% sum of axis and the motion of rotation would be changed; the seas, leavthe fractions; and 1322+4=1326% the sum required. Otherwise : Given mixed numbers. ing their ancient beds, would spread themselves over the new equator; a great part of the human race and the animal tribes would be either drowned in this universal deluge or destroyed by the violent concussion impressed on the terrestrial globe; all the monuments of human industry would be demolished; such are the disasters which the shock of a comet would produce. Thus we see how the ocean has formerly covered the highest mountains, on which it has left indubitable marks of its presence; we see how the animals and plants of the south may have formerly existed in the climates of the north, where their remains and their impressions are still found; lastly, we see how the newness of the moral world may be accounted for, its monuments not reaching beyond 5,000* 212 261 sum * More than 7,000 according to the Septuagint. 11109 Whence, 1912 1221326 sum required. 3945 137988 and it passed its perihelion in 1838, &c. There are still other two comets of which the course is supposed to be known; but as their returns have not been verified, we must wait till observation has confirmed these expectations. The first is the comet of 1680, which Newton made the subject of his researches, and to which he attributed a revolution of 575 years; this would be, according to him, the comet which appeared in 1106, 531, and in the years 34, and 619, before the Christian era; the second is that which was seen in 1264, and 1556, to which a revolution of 292 years is attributed. Of all the known comets, that of 1772 approached nearest to the earth; it appeared so rapid in its course that it described in one day 120o or two-thirds of the arc of the visible hemisphere of the heavens, retrograding through the signs of the Virgin, the Lion, and Cancer. The comet of 1770, calculated by Lexell, was only distant from the earth by six times the distance of the moon, or about one million and four thousand British miles; its distance from the sun was only about part of the distance of the earth: hence it went as hear to the sun as it did to the earth. This comet has only continued to torment and perplex modern astronomers who have attempted to discover its course and its motions. THE POPULAR EDUCATOR. years. The human race thus reduced to a small number of indi- | approach. The comet of 1770 which was very near the earth, ocviduals, and to the most deplorable state, were uniformly occupied casioned no disturbance in its motion. Laplace has calculated that for a very long time with the care of providing a subsistence, and if its muss had been equal to that of the earth the action of this comet they must have entirely lost the remembrance of the sciences and would have increased the sidereal year by 2 h. 28 m. Its mass was the arts; and when the progress of civilisation made them feel certainly not the 30th part of that of our globe, and probably much anew the want of them, it was necessary to recommence the whole, less, since it passed Jupiter and his satellites without occasioning as if the race had been newly placed upon the earth. Whatever the least perturbation in their motion. The masses of the comets cause may be assigned by some philosophers for these phenomena, are, therefore, very small and are far from altering the motion of the we may be perfectly secure respecting such a terrible event, during planets; but they, themselves, must greatly experience the influence the short interval of life. But man is so disposed to submit to the of the latter. In consequence of this influence some may have their impression of fear, that in 1773, the most striking alarm spread elliptic orbits changed into parabolic or even hyperbolic orbits; over Paris, and was thence communicated to the whole of France, these moving then in open curves infinitely extended, must on the simple announcement of a memoir, in which Lalande had depart from our system never to return; and, leaving the sun at all determined such of the observed comets as might approach the distances, must enter into the sphere of attraction of some other earth, the nearest; so true is it that error, superstition, needless star in order to become a satellite or to describe a new paraalarm, and all the evils which ignorance brings in her train, are bola; thus incessantly changing the focus of attraction, alterreadily brought into action, when the light of science is extin- nately approaching to one or receding from another, until at guished." If it be true, as is supposed, that the comet of Newton last they are perhaps precipitated towards some central or attracting accomplished seven revolutions in 4,028 years, it must have passed body, where they are either shivered to atoms or absorbed into the near the earth 2,349 years before the Christian era, about the year mass. when the Mosaic deluge is related to have happened, according to the Hebrew text of the Scriptures. The celebrated Halley was the first who supposed that the revolutions of our globe, which are attested by historical facts, and by the observed state of its surface, were produced by the shock of some comet. In America, an immense body has been seen passing with great rapidity through the atmosphere or space which surrounds the earth; if we suppose this moveable body or projectile to be a mass 2,000 times less than the earth itself, and to have the density of granite, it would form a globe of about 600 miles in diameter; if this body had struck the earth with a velocity of forty-five miles per second, M. Olbers has calculated that the shock would have reduced our globe to shivers, as perhaps has been the case with the twenty new planets, which we have supposed formerly to be one, a conjecture made by this astronomer when only three or four were discovered. Aerolites or meteoric stones are supposed to have had a similar origin; some of them have been found of an immense size. That described in the "Philosophical Transactions," vol. 6, was estimated at 120 millions of quintals or about five millions of tons; some of its fragments only were thrown upon the earth; its velocity was about nineteen miles per second; and it passed the earth within a distance of about twenty-two miles. If it had struck the earth, terrible effects would have been the result; a general earthquake and a change in the axis of rotation, or in the place of the poles, would have been among the consequences. It is not impossible, therefore, that aerolites may be non-luminous bodies belonging to the solar system, moving in space according to the laws of gravitation, and which become only visible when they fall to the earth. We are only acquainted with the nature of aerolites, after they reach us, when they have traversed the atmosphere, where probably they are formed in a state of incandescence, or perhaps even of fusion; for some have been seen which have evidently undergone this process, and were even burning at the moment of their fall. Of 240 of these meteoric stones examined by M. Chladni, some have appeared as tails, belts, or streams of light, which were collected into a globe of fire; their rapid course has been observed, and their explosions have been heard. Falling stars are, perhaps, only meteoric fragments, luminous and wandering points, a species of small comets rendered visible, while traversing our atmosphere, by the prodigious heat which they develope in their extremely rapid course. The bulk of many aerolites has exceeded that of some of the new planets; for it has been calculated that Ceres is only 200 miles in diameter. The meteoric stone, which fell in Calabria, in March, 1813, was accompanied with such extraordinary circumstances as have led philosophers to suppose that such stones are the fragments of some cometary mass. "In fact," says M. Francoeur, "there has been seen coming from the sea, from the eastern quarter, a red cloud which spread darkness and alarm; there have been heard in the air a tremendous noise and a groaning like that of the angry deep; there have been seen flashes and trains of fire; and there have fallen huge drops of water, masses of stones and red sand, which have been scattered all around. This sand was composed of clay, lime, iron, and chrome." Such are the phenomena which accompany the fall of meteoric stones. Many philosophers suppose that comets have only a small quantity of matter, and that they are only a species of condensed vapour; as an evidence of this, astronomers have hitherto observed that no derangement has been caused in our system, even by their nearest 330 Thus the comet of 1770 observed in so large an arc of the heavens as 170° ought, according to the calculations of Lexell, to reappear every five years and a half; for his calculations were made and repeated again and again with care, and yet it has never been again observed, and never returned since its appearance that year. There is no doubt that the attraction of Jupiter and Saturn, between which its aphelion point was situated, had at first wheeled it round to us, as it were, to render it visible in 1770, the probability being that it was invisible before; but nine years after this, these same planets by a contrary action again altered the elements of the orbit, and rendered the comet invisible to us for ever, or at least impossible to be recognised as the same. The substance of which the comets are composed is unknown, and there is only one probable supposition hitherto mentioned respecting the matter of the tails by which they are accompanied. In 1811, M. Chladni observed a prodigious ebullition in the comet of that year; the undulation or motion produced by it was trans. mitted in two or three seconds, from the nucleus to the end of the tail, a distance of about twelve millions of miles. This wonderful velocity surpasses our conception; it is even greater than the velocity of light. The tail of the comet of 1807 had exhibited the same pheno mena to M. Chladni; so that there can be little reason to doubt the fact. The ancients generally believed on the authority of Aristotle, thet comets were fiery meteors, composed of vapours emanating from the earth, and which were condensed in the air. This opinion now-a-days, could not sustain examination; we are certain that comets are conical bodies moving, like the planets, in consequence of an original in pulse, and of the attraction of the sun. But if we are asked what is the nature of the substance of which they are formed, we must confess our total ignorance. Thus the indefatigable researches of man, the waves of the troubled sea, have their limits beyond which they cannot pass; and more than once, in the course of his investi gations, has it been said to him, "hitherto shalt thou go, and no fur ther; and here shall thy proud triumphs be stayed." It is one con solation, however, that the knowledge of the composition of these bodies is rather a branch of the boundless science of natural philosophy, than of astronomy, and that the triumphs of the latter science remain unaffected by our ignorance of such a fact, and irreproachable on its account. Comets are not seen for a longer period than six months together. Those which were visible for the longest period, as we learn from history, were the comet which appeared in 64, in the reign of Nero; that in 603, in the time of the impostor Mahomet; that in 1240, at the period of the irruption of Tamerlane; that in 1729; and that which some of our readers will remember, the comet of 1811. The number of the comets is unknown; of 130 which have been observed in the last ten centuries, seventy-two have been observed since the invention of the telescope. There is no doubt that a much greater number of those bodies has escaped observation. Only two were seen in the thirteenth century, and two in the fourteenth, because the attention of mankind was only turned to those which were rendered remarkable by a large disk or a long tail. Some astronomers conjecture (we think rather gratuitously) that there are, perhaps, 250 millions of comets which never approach nearer to the sun than Uranus, and on which we can never make any observation. We have seen that in former ages a baleful influence was attributed to the appearance of comets. Science has at last disst pated these imaginary terrors; and in proportion as events were discovered not to correspond with the predictions of the ignorant or the designing, the fears produced by their appearance have ceased. The comet of 1664, first seen in the constellation called the Crow, was to have caused the death of all crowned heads; and yet that year passed away without any death of the kind! The two Comets of 1402, which seemed to be ornamented with tufts of feathers, were attended with no other misfortune than the death of a duke of Milan; and those which recently appeared in the midst of so many political convulsions, have not, thanks to the march of intellect, been accused of having produced any of the untoward events that have occured. We have even a delightful recollection of the abundant crops which accompanied the appearance of the Comet of 1811! Many of the daily papers in France having asserted, no doubt on the authority of some astronomers in that country, that one of the recently observed comets would impinge on the earth and break it to pieces; the French Board of Longitude thought it advisable to give a statement of all that was known with certainty on the subject, in a memoir written by M. Arago, one of that learned body. From the laborious calculations of M. Damoiseau, another member of the same body, it was announced:-19. That in the middle of November, 1835, we should see again, passing near the sun, that comet which in 1455, with a tail of 60° in length, terrified all Europe by its brilliancy, by its giving plentiful birth to astronomical predictions, and by a superstitious application of its appearance to the fearful progress of the arms of the Mahometans. 2o. That the comet of Encké should reach its perihelion on the 4th of May, 1832; but that the astronomers of the Cape of Good Hope and New Holland should be much more advantageously situated for observing it than those of Europe. 3°. That the comet of Biela, whose period is six years and three quarters, is that which was announced as threatening our globe with such dreadful ruin during the year 1832! Concerning it M. Damoiseau determined by long, minute, and laborious calculations, that it would cross the plane of the ecliptic, that is, the plane in which the earth moves, on the 20th of October, 1832, before midnight. Now, as the earth during its course round the sun never quits the plane of the ecliptic, it is in that plane alone that a comet can strike against it; so that whatever dangers were to be feared from this comet should have happened on the 29th of October, 1832, before midnight! M. Arago, in the memoir referred to, proceeded to argue from the possibility of a change in the elements of the comet before it reached the said point, that no mischief could happen to the world on the night in question. Independently however, of the uncertainty attending all astronomical calcnlations regarding the course of comets, a theory still far from being complete, we might have rested in the assurance that no catastrophe could overtake the globe on which we are placed, till the propheeies of Scripture and the decrees, the revealed decrees, of the Almighty were fulfilled. Glossary of Terms used in the preceding articles on Comets. Aphelion, that point in the orbit or path of a comet, or planet, the most distant from the sun. Arc, any portion of a circle; such as the visible arc of the heavens, which is a semicircle. Aris, the diameter of a sphere, on which it revolves, &c. Constellation, a collection of bright stars in the heavens, arranged according to the practice of astronomers. Disc, the face of a comet or planet which appears to us enlightened Eclipse, the darkening of a luminous body by the intervention of an opaque body between it and the eye; as an eclipse of the sun, by the intervention of the moon. • Ecliptic,the plane of the orbit or path of the earth round the sun. Ellipse, an oval curve, or path of a comet or planet, explained at p. 225, ccl. 2. vol. II. Elongatio, a stretching out, extending, or making longer, any distance. Focus, the central point of a curve, or path of a comet or planet, in which the attracting body is placed. Foci, two points in the ellipse equally central. Hyperbola, an open curve somewhat like a parabola explained at p. 226., col. 1, vol. II. Meteor, an unusual body which appears in the atmosphere. Nodes, the points where the orbit or path of one comet or planet intersects the plane of another. Nucleus, the interior of a comet which is considered to be more solid than its exterior. Opaque, non-transparent; what cannot be seen through. SOLUTIONS TO PROBLEMS AND QUERIES. We have so often been requested to answer queries similar to the following, that we proceed to give a general solution which will be applicable to all. "At what time will the hands of a clock or watch be together between four and five o'clock ?" The time between the meeting or conjunction of the hour and minute hands of a clock may be likened to the conjunction of two planets moving in different orbits. Suppose that the two are in conjunction at any given instant, for instance, the hour and minute hands are in conjunction at noon, then, after any elapsed time, the hour-hand will have described an angle m, and the minute-hand an angle M, round the centre of the dial. Now M is greater than m, therefore, at the end of a given time, say one hour, the separation of the minute-land from the hour-hand (measuring this separation by an angle formed by the two hands, considered as straight lines, at the centre of the dial,) will be M m; at the end of two hours, the angle of separation will be 2 (Mm); at the end of three hours, it will be 3 (M = m); and, at the end of t hours, it will be t'Mm). When the angle of separation is 360°, that is, when t (M — m) = 360°, then there will be another conjunc tion. Whence, 1= 360° M m In this equation t denotes the time between one conjunction and another, M and m denoting the hourly motions of the two hands. Let p and P denote the times employed by the minute and the hour-hands, respectively, to complete one entire revolution of the dial; then, because 1 hour: Phours:: M° 360°, and 1 hour: p hours :: m° 360°; we have M = 360° and m = 360° ; whence = 360° Pp Number of Conjunctions. 360° (p-P Substituting, in this formula, the values of P and p, which are 1 hour and 12 hours respectively, we have t= = 1 X 12 12 = 1 hour. 12 Î1 Whence, the time of a complete conjunction is one hour and one-eleventh of an hour; of two complete conjunctions, twó hours and two-elevenths of an hour, and so on; consequently. the time of n complete conjunctions is n X 1. The following table shows the number of conjunctions which happen after noon, and between noon and midnight, and the times of each. 1234567 8 10 11 21 27 32 38 4371 49TY 54 midnight. " 234567890 10 |