spicuous. Stars further to the east, and holding a more elevated position, mostly pertain to the constellation Draco, or the Dragon, coiling at apparently a short distance from the Pole-star; and nearly due east is the star Etanim, belonging to this constellation. Westward, or on the lefthand, and almost opposite to Etanim, are bright stars composing the constellation Auriga among these Capella and Alajoth shine conspicuous; the former appears nearly W. by S. from the Pole-star, at a considerable elevation; with Menkalina, pertaining also to Auriga, and beaming eastward. The chief stars connected with the Great Bear, although figured as guides among the starry glitter, are too well known to render any further description necessary. Such are the prominent constellations which diversify the northern part of the heavens about the beginning of this pleasant month, when stars and flowers delight equally the astronomer and, botanist. There are also several brilliant stars of the first magnitude which require notice. Look, therefore, E. by S., and the beauteous star Arcturus may be seen in the constellation Boötes, about midway between the horizon and the zenith; north-east appears a star of not less beauty-Vega or Lyra, elevated 20° above the horizon, in a direction nearly opposite Capella. Deneb, pertaining to the Swan, shines farther north, and at a lower elevation than the Lyre. Midway between the western horizon and the zenith, yet verging to the west, Castor and Pollux recal the thoughts to a period of remote antiquity. Further down, and nearly on the horizon, almost due west, are Betelgeux and Bellatrix, in the shoulders of Orion, who has partially descended below the horizon. South-west, and midway between Pollux and the horizon, gleams Procyon, a star of the first magnitude, in Canis Minor, or the Lesser Dog. With reference to such among the heavenly luminaries as are depicted in the circle, it must be held in mind that they perpetually visible, leading a circling dance above the horizon, and having the Pole-star as their centre. are seemingly above the pole, and even near the zenith; at another, even below that point, and verging near the northern horizon. When riding high, they appear to move from east to west; when otherwise, from west to east. Such as are near the Pole-star describe small circles-such as take a wider range necessarily describe larger ones; but their periods of apparent revolution are the same, that is, 23h. 56m. 4 sec. It is likewise desirable to remember, that as the observer is supposed to be in 52° north latitude, all stars within 52° of the pole never descend below the horizon, varying as the weeks roll on; at one time Such stars as we have thought it desirable to represent, are prominent, and obvious in the heavens; others might have been introduced, but they would tend to perplex beginners; and in order most readily to estimate apparent distances between the stars, as also from the horizon, it will be found a great assistance to bear in mind, that the space between the two pointers comprises exactly 5°; and between Dubhe, nearest to the pole, and the Polestar, 29°. True it is that the pointers come to the southern meridian, or are nearly at zenith at ten P.M., about the beginning of April; but not till the 7th of the month do they occupy the exact position above named, and this takes place at 10 in the evening. We recommend our readers to make a pasteboard circle, as the one we have pictured, though much larger, and copy the stars thereon; to obtain, also, a square board, or piece of thick pasteboard, and form a larger circle of months and days, the lesser circle being affixed to the larger, about the centre, so as to move readily; an astronomical clock may be thus obtained, for pointing out the hours of the night, and showing the positions of the circumpolar stars at any hour of the day or night. The idea is suggested by a most ingenious planetarium, to which we are indebted for much instruction. Our two last papers spoke concerning the mythological history of some distinguished constellations; we shall now refer to the phenomena of double stars, in each, as discovered by the aid of telescopes, premising our remarks by saying, that little was known with regard to this interesting subject till Sir William Herschel commenced his extensive observations on the sidereal heavens. Astronomers were, indeed, previously aware that double stars held an interesting place among their brethren, but they had not extended their discoveries; they contented themselves with ascertaining their existence, and noting six or eight in their charts. When, however, the thoughts of Sir William Herschel were directed to the subject, and his telescope swept the starry heavens, no less than 500 double stars were added, having their situations and relative positions distinctly marked. The son of this illustrious astronomer discovered many more: his unwearied labours, with those of Sir James South, produced an additional list of 380; subsequently, Sir J. Herschel formed a distinct catalogue of 3,300 double and even triple stars, the result of his own observations, accompanied with precise measurements of their distances and angles of position; and Sir James South identified 480, the result also of his own labours. Since then, further discoveries have increased their numbers. The celebrated astronomer, Struve, makes mention of no fewer than 3,000 double stars; in the progress of identifying which, he examined about 120,000 of those sparkling luminaries which gem the vault of heaven. The southern hemisphere reveals 250 stars of the same description-according to the testimony of Mr. Dunlop; and during a late residence at the Cape of Good Hope, Sir James Herschel added considerably to their list; it is, therefore, conjectured, that 6,000 have been made the subject of accurate research. Connected with the mention of double stars, is that important discovery, which realises the fact of a progressive and regular change, bearing in some stars chiefly on their position, in others on their distance, and which results from the smaller star revolving round the larger in an elliptical or circular orbit, although occasionally both stars revolve around some central point. Those who narrowly observe the heavens by aid of a high magnifier, may readily ascertain the fact of a revolving motion in such stars as are called double. At one time the satellite or smaller star disappears, in consequence of becoming obscured while passing behind the other; as Jupiter or Venus is occasionally invisible, when on the opposite side of the sun; or the satellites of Jupiter, if similarly circumstanced with regard to that planet. Three stars have occasionally been seen revolving about a common centre, and even four or five. The orbits in which one star circles round another are generally elliptical, similar to the path described by the earth and other planets when revolving round the sun; as also those in which the satellites of Jupiter, Saturn, and Uranus perform their revolutions. These orbicular motions are either retrograde or direct, or in the same direction as the movements of our own planets; and very curious is the fact with regard to the star Serpentarii, in common with many others, that the revolving star appears to move in a straight line, and to oscillate on either side of the larger star, around which it revolves in a manner similar to the satellites of Jupiter, which pass apparently from one side to the other of the planet in nearly straight lines-an effect resulting from the plane of their orbits being nearly in a line with the eye of the observer. When Sir William Herschel first directed his attention to the subject of double stars, the two stars to which we have referred were distinctly separate; at the present time the lesser star is so completely projected on the other that even the most powerful telescope cannot reveal any separation-and why? Because one star is passing across the disk of the other, and will not again be visible till after the lapse of many years. In our next we shall instance several binary, or double stars, belonging to such of the constellations as have already been regarded in a legendary or historic point of view. Taurus, one of the zodiacal constellations, pertains to this month, and may readily be discovered near Perseus and Orion. It is needful to remark, that the sign or figure, as represented in the earliest charts, and retained to the present day, is merely the front portion of the animal. The sun passes from the sign Aries to that of the Bull, on the 20th, at 5h. 3m. P.M. The moon is in the constellation Pisces till the 3rd, when she enters Cetus; Aries claims her on that day, Taurus on the 4th, Orion on the 7th, Gemini on the 8th, Cancer on the 9th, Leo on the 11th, Virgo on the 13th, Libra on the 15th, Scorpio on the 18th, Sagittarius on the 23rd, Aquarius on the 25th, Pisces on the 27th, Cetus again on the 30th, Aries on the 31st. Venus is in the constellation Aquarius till the 16th; in Pisces from the 17th to the end of the month. Mars occupies a place in Aquarius till the 6th; in Pisces from the 7th to the end of the month. Both are morning stars. MATHEMATICAL QUESTIONS, SOLUTIONS, &c. EDITED BY W. J. REYNOLDS, ESQ. B.A. Solutions to Questions on Pp. 91, 121, & 152. 17. We will first determine what could have been the gain by the sale of one apple, supposing 5 per cent. to be cleared on the outlay. Dividing the whole gain by this (both being expressed, of course, in one denomination), we shall evidently find the number of apples sold. Now, since 5 apples were sold for one penny, 106 times 5, or 530, were sold for 106 pence; and therefore this number of apples was bought for 100 pence, as 6 per cent. was gained by selling them at the above rate. Therefore, ex æquo, : OG OL: ME MF. (Euclid, B. v. Again, by the similar triangles KOD, OK OD :: ME: MD. And by the similar triangles HOD, OD: OH :: MD: MF. OK: OH :: ME: MF. But, from above, OG OL:: ME: MF. OG: OL:: OK: OH. (Euclid, B. v., Q.E.D. Therefore the rectangle OG, OH = the Consequently, had the fruit been dis-rectangle OK, OL. (B. vi. Prop. 16.)— posed of so as to gain 5 per cent. on the outlay, 530 apples would have gained 5 pence, and therefore one apple would have gained, or 16, of a penny. And since, by the question, the whole gain at this rate would have been 3s. 1d., or 37 pence; the whole number of apples bought is 37 106. And 376 35 10675 × 53 3975. Ans. 19. The demonstration of this theorem may be made to depend upon that of the bers:-"The difference between any given following well-known property of numnumber and the number expressing the sum of its digits, is exactly divisible by 9." And to prove this, a few simple considerations may be adduced. In the first place, the difference in ques 18. Let CD and EF, produced if neces- tion will be the same, whatever the figure sary, meet in the point M. M H K A G L E B Then, by the similar triangles GOC, EMC, : OG OC: ME: MC. (Euclid, B. vi. occupying the unit's place may be, and therefore the same as if it were a cipher. For, by substituting a cipher for the unit's figure, we diminish both the number and the sum of its digits by the same quantity, and therefore the difference between the two will be unaltered. Again, the figure in the second place (reckoning from left to right) represents a certain number of tens, and if from this we take as many ones, the difference must be the same number of nines. Similarly, the figure in the third place represents a certain number of hundreds, and if from this we take as many ones, the difference is the same number of ninety-nines, that is, an exact number of nines. The same process applied to the fourth place will give a number of nine And by the similar triangles LOC, hundred and ninety-nines, which is also an FMC, OC OL:: MC: MF. exact number of nines. Continuing this operation, it is evident that we shall at ..... last have taken away from the given num-+ Referring now to the theorem proposed, it is clear that the ultimate result will be the same whether we add all the numbers together, and from the sum so obtained subtract the sum of the digits; or subtract from each the sum of its digits in the first place, and then add the several remainders together. But by what has been proved above, these several remainders are all exactly multiples of nine, and consequently their sum will evidently be exactly divisible by 9. Hence the truth of the theorem. a (10-11) + b2 (10-11) +c, (10-11) + . . . + a2 (10~~2 — 1) + b2 (10×1—2' — 1) +c2 (10~2~2 — 1) + + ... Now, as ak. 1 (k being a whole num1, it ber) is always exactly divisible by xfollows that 10-1 is always divisible by 10-1, or 9. Consequently each term of the above expression for the difference is divisible by 9, and therefore the whole expression is so divisible, and the property in question is proved. Several correspondents appear to have considered this problem, and some analogous ones, with great attention; and this circumstance suggested to the Editor, while engaged in examining the solutions transmitted to him, that a beautiful property of numbers (which, if we remember rightly, was given by Mr. Adams, the celebrated physical astronomer) would be interesting to many mathematical readers He has therefore of the Family Tutor. proposed it below, as the 27th Question for Solution. QUESTIONS FOR SOLUTION. 27. If the odd numbers be arranged in order of magnitude, and then, beginning at the lowest, separated into groups, the +b1-1 10 first group containing one number, the second two, and so on-prove that the sum of the numbers forming any group will be +C2-110 expressed by the cube of the natural number which marks the order of the group. 28. A contractor having undertaken to make a road 12 miles 800 yards in length in 5 months, sets on at first 92 workmen, and at the end of 3 months 4 miles 320 yards of the road are completed. How many additional men must he set on in order that he may complete the work according to the contract? 136-Luminous Metal. A. H.-The luminosity of a freshly-cut surface of potassium was discovered by Mr. W. Petrie. The experiments proving the fact are recorded in the Proceedings of the British Association, 1850. 137-By Hook or by Crook.-D. J. M.-It is said that Strongbow, when debating with his followers on the best mode of capturing Ireland said, "It must be taken by hook or by crook. The N.E. boundary of Waterford harbour is known as "The Hook;" and "Crook Haven" is an equally well known harbour on the south coast: hence the words of the beseiger. 138-Esquire. D. O. This title is now given to every man of respectability; but persons entitled to superior consideration are distinguished by &c. &c. &c. added to the superscription; and it is deemed more respectful to write the word Esquire at full length. It is customary to add the initials of professional rank after the ordinary title, thus:-Charles Cowan, Esquire, M. D., W. J. Reynolds, Esquire, B. A., &c. &c. 139-Mild Winters. T. A.-Our pupil asks for the "reasons adduced by scientific men for the comparative warmth of our present winters to those of which we have heard our fathers speak." The answer to the question it is difficult to give. By many the phenomenon has been attributed to the precession of the equinoxes. It is expected that when the cycle of precession has been completed, a succession of severe winters may be experienced. 140-Queen Boadicea. M. T.-There was formerly a church in Oxford dedicated to her, and called St. Budoc. The name Budic is sometimes to be found among the Welsh, and was once common in England. In Cornwall there is to this day a parish named St. Budoc, and the name of Boduc and Budoe on the ancient British coin, is generally supposed to be intended for the famous Queen Boadicea. Dio writes her name in Greek, "Boudouica." 141-Head Money. H. K.-Before the Conquest a regular scale of prices was established, which persons who caused the deaths of others should pay for the offence. The price of the king's head was settled by Edmund at about £13.900 of our present money; a prince, one-half that sum; a bishop or alderman, one-fourth; and the lower order of freemen, a few shillings only. Edmund also ordained that no criminal should be given up from the sanctuary, or place of refuge. 142-Botanical. Fire of London. D. R.-The plant which sprung up so abundantly after the Great Fire of London, was the Sysymbrium iris, or hedge-mustard. So profuse was the plant, and so quickly did it cover the ground where the consumed city had stood, that it was supposed by the hotanists of those days, that a greater quantity existed on that one spot than could have been collected from the whole surface of Europe. Such a singular instance of vegetable growth no naturalist has yet been able to account for. 143-Latitude. W. P.-Latitude is not east and west, but north and south, of the equator. East and West is longitude-either east or west of Greenwich meridian line. The method of finding the latitude on board ship is by ascertaining the true meridian altitude of the sun at 12 noon by a quadrant, and then by subtracting this from 90 deg., thus obtaining the sun's zenith distance; from this the sun's declination (taken from the Nautical Almanack) of each day is deducted, and the result is the latitude. The method of ascertaining the longitude requires more complex calculation; for this we must refer our pupils to Norie's Epitome on Navigation. con speech, with all the various topics to which it might refer, For some 146-Letter Writing. J. T.-It is no part of the Tutor's plan to "publish a complete Letter-writer." Such guides do more harm than good, by teaching persons to write artificially. The great art of letter-writing for purposes of ordinary correspondence is simply to write grammatically, in plain and courteous terms, what the writer would have said if the person addressed had been sufficiently near for oral communication. Letters should bear the name of the street and place from which they are sent, and should for business purposes be as brief as possible. It is the fashion to commence a letter with Sir, "-"Dear Sir," or "My dear Sir,'-according to the intimacy existing between the parties; "Sir" is sufficient for all except private and friendly letters. Communications containing orders for goods should always be signed in full and legible characters by the writer, prefaced by such terms as Your obedient servant,"-" Yours obediently," "Yours, &c.," or "Yours truly." persons on going to meet a friend would ask another what they should say to him, but this is scarcely less absurd than the habit of conducting correspondence by stereotyped letters of ridiculous formality, [See Family Friend, vol. i. p. iii.] 147-Railways. Few C. R.-The first propounder in print of a general system of iron railways, with steam applied as a motive power, was Thomas Gray, in 1820. We met with a copy of his pamphlet in Manchester a few years ago, and were much struck with the completeness of the plans of a man whose name was comparatively unknown, but to whom the credit of the invention of railways appears to belong. Gray, like other enthusiasts, who seem to have come before their time, lived neglected and died poor; while men around him became millionnaires by buying and selling shares in the very line he had first agitated, viz., that between Liverpool and Manchester. About four years ago, it was proposed to erect a public monument to his memory, and miniature plaster casts of his bust were circulated, with statements of his claims; but the agitation excited no attention, and the project was soon abandoned. George Stephenson differed from Thomas Gray in his plans, and was an inventor in the large sense of the word. Gray proposed slow travelling and heavy loads; while Stephenson, by a combination of new powers and materials, produced an iron road over a quaking moss, along which a fire-steed travelled, "Chasing the flying bird," at the rate of half a mile in a minute. We have not the materials for a biography of Thomas Gray, but we hope that those who possess them will give them the publicity which they deserve. 148-Music. J. V. N.-It is possible to learn to play upon any musical instrument without master, if a good instruction book is obtained; but the cheapest method of gaining practical knowledge of any instrument, is to engage a competent master, and diligently to practise according to his directions. We say cheapest, because so much time. the most valuable of all things, is usually mis144-Destructive Influence of Tides. U. S.-The de- spent in musical if-education. It is impossible for us structive influence of the high tides has been long a to give a list of prices at which good instruments ean subject of consideration by scientific men. At the meeting be obtained. The Tutor takes this opportunity of cautionof the British Association held at Edinburgh (July 31-ing his pupils of both sexes against hastily commencing to August 7), Mr. Stevenson reported the result of certain learn to play upon musical instruments. He has met with observations made by him on the force of the waves with only one man who in youth had not attempted to learn to reference to the construction of marine works. A self- play the flute and yet how very rarely do we meet with registering instrument, consisting of a disc on which the Hlute players. Every youth begins, gives a great deal of sea infringed, was used. The force of the wave was time, and often bestows an amount of perseverance worthy measured by means of a spiral spring. It appeared from of a better pursuit to the study of the instrument, and the experiments, that the force of the tide in the German finds out, perhaps, at last, that he will never attain even Ocean is about 1 ton per square foot, and 3 tons for the to mediocrity as an amateur performer; so the flute is thrown by in disgust. Taking all things into consideration, the piano or the organ afford the greatest opportunities of pleasing with comparatively little labour; but to learn to play upon either of these instruments will occupy a very considerable time, and require persevering labour. It is the duty of parents and the superintendents of schools to discourage the desire for the study of music in adults, who are animated by the desire of playing for the sake of pleasing or attracting admiration, merely; and only to afford facilities to the pupils in whom a decidedly musical taste is exhibited. The genius for music may be developed or cultivated in the child; but the want of Atlantic. 145-Reporting. T. M.-The phonetic short-hand, invented by Mr. Pitman, and perfected for reporting by Mr. Reed, is undoubtedly the best and most philosophical system. It is almost impossible to give advice upon the detailed education of a boy intended as a reporter for the press. To report well, it is always desirable that the short-hand writer should be acquainted with the subject on which the speaker descants; and therefore to report everthing well, it may be supposed that it is necessary to know something of everything. For example, a short-hand writer who could give a verbatim report of a political |