without contingency. Shall supplies the place of either second person is dependent upon the will of the (any one) of these signs, when the future involves a doubt, or a contingency, especially when it regards the first, the first person says to the second, thou shalt, first person. Its comparative power is simple, and is and not thou wilt, for the willing rests with the never abused; its contingent force alone creates the difficulty. With respect to human life and purpose, doubt first; but if the first leaves the second to act as he is the very essence of futurity. The vivacity of the may think proper, he says thou wilt, and thus southern nations confounds will and h, because they claims no control over that willing. Again; in determine, in the levity of their minds, without doubt or dread. The cautious Englishman doubts ever; he sees the third person, he shall or he will, we still see contingency in the future; and from this peculiarity of the same principle. When the first says he shall, of the national mind comes a delicacy of expression which has no equivalent in any other language.' he deprives the third of the exercise of his own "This is rather a moral than a grammatical dis-will; but when he says he will, he leaves him the quisition and leaves the subject very much in the exercise of that will, and simply expresses his belief that it is the intention or will of the third persame state as we found it. son to do this or that. "Though the auxiliaries shall and will, in certain positions, are very different in their meaning, yet they are frequently used the one for the other, as exemplified in the trite but forcible case of the drowning Irishman; "I will drown and nobody shall help me;" and it is a common expression used by an Irish servant, “Shall you take tea tonight, and will I bring it in?" The Scotch are also apt to confound the use of shall and will; as, Without having attended to this, we will be at a loss, in understanding several passages in the classics, which relate to public speaking and the theatrical entertain ments of the ancients.'-Blair's Lectures. "The principle to be borne in mind in the distinction of shall and will is, that it is always the first person that speaks; and that, in the first person of the verb, the speaker is also the agent; but that in the second and third, the first person is the speaker, but the second or third the agent. Hence, as volition is allowed to the second or third person, or control exercised over that volition, the use of will and shall must varv in passing from the first to the second and third persons. When, however, we pass to the interrogative forms of shall and will, the case is reversed. In In the Latin language there are no two words we the second person of the verb, we simply inquire would more readily take to be synonymous than amare what the will of that person is, the act of volition and diligere.'-Ibid. remaining in the breast of that person, and not Shall and should are required. Think what reflection shall most probably arise.'-being subject to the control of the person asking. Ibid. There is not a girl in town, but let her, in going to a mask, and she shal dress as a shepherdess '-Spectator. Will. We, therefore, say wilt thou? or will you? In the third person of the verb again, the act of willing remains with that person, and we simply ask for an enunciation of that will; as, will he, if plural, will they? Thus; Shall I go to London? wilt thou go? will he go? shall we go? will you go? will they go? "This distinction of shall and will does not form “A well-educated Englishman, however, seldom a part of the system of a Latin or Greek verb. makes a mistake in the application of shall and "The verb will is also a principal, as well as an will, though it may sometimes be necessary to feel auxiliary, As, I will, thou willest, he wills, we will, the way, as it were, by a delicate touch. There ye or you will, they will. A confusion of will as must be, and is, a broad principle of distinction. a principal and as an auxiliary occurs in the folIn the modest language of the reviewer just re-lowing passage from Atterbury's Sermons :' ferred to, let us try to find it. From example, let us endeavor to work out the principle. Thou that art the author and bestower of life canst doubtless restore it also if thou will'st, and when thou will'st; but whether thou will'st please to restore it, or not, thou alone knowest.' "Here will'st in the two former cases is a principal, in the last an auxiliary, and ought to be wilt, and not will'st or willest; that is, but whether thou wilt restore them or not, &c." "I shall go to town to-morrow. Here simply the intention of doing a certain thing is expressed, without any anticipation of, or reference to, hindrance. But when I say I will go to town to-morrow, I declare my resolution to do so, in spite of ail opposition. I must and will go to town to-mor[We merely append the following examples of row. Now, we must bear in mind that, in both the use of these words, with the meaning as we these cases, the person that speaks is also the per. understand it, according to the views above exson that is about to act. He, therefore, at pleasure, expresses an act of simple volition, or of fixed purpose, according to circumstances. Both are at his own option; he has the control of both in his own mind. But when we pass to the second person, thou shalt or wilt, it is to be borne in mind that, though the second person is the actor, the first is still the speaker, lf, therefore, the acting of the pressed.-ED. Denoting will or determination on the part of the speaker: I will write, You will write, He shall write, We will write, Denoting a simple prediction; ture event; I shall write, merely foretelling a fu We shall write, Solution of the Horse-Round-the-Tower Problem, in the September Number. "If a horse be tied to a circular tower by a rope wound round-and-round the tower, required to know: (1.) What distance would he travel, beginning at the tower, and keeping the rope tense in unwinding, in one, or any given number of revolutions, the diameter of the tower being 20 feet? (2.) The area between the curves thus described and the rope, or radius, at any time? (3.) The length of rope (less than the semi-circumference,) in order that the horse may feed over 1502.58083 square feet?” E To find C, make x = 0,... y = 0, then C = 0; 2a At one revolution becomes == AE circumference of circle = 2a; . ... y=2a2 ABDE; that is, the first involute is equal to the circumference of a circle whose diameter is the circumference of the evolute ! Putting n number of revolutions or involutes, whole or fractional, and c = circumference of evolute, x = nc, ... y = n2c', letting c represent the first involute. In this example = el 20 × (3.141592) 197.3919659 feet. The whole curve for one, two, three, four, &c., revolutions will be respectively, el, 4el, 9c', 16c', &c. But the lengths of the separate, successive involutes, first, second, third, fourth, &c., will be respectively, c', 3c', 5c!, 7el, &c.; and generally, any involute nn2c! — (n − 1)2c! = (2n − 1)c'. (2.) To find the area. 2a Let the circle AIT represent the horizontal pro- Integrating, A = — jection of the tower, from which the rope may be conceived to unwind, beginning at A. The end of formula, there being no the rope will describe the curve ABDE &c. The circle AIT is called the evolute; the curve ABDE is called the involute, evolute and involute being correlative terms; the rope unwound at any time, feet, the area to AE. Making = c = AE, 23 2a as IB, TD, or AE, is called the radius of curva- the general formula only in a particular sense, viz: it expresses the area described by the radius of Draw any radius IB, which will be equal to arc Al, and denote it by x; also denote the arc of involute AB by y. Also draw a consecutive expressions for the areas described by the radius radius TD. The arc IT will be de; and the arc of curvature in 1, 2, 3, 4, ....n revolutions respecBD, the corresponding increment of the function tively. Hence to find the area described by the y, will be dy. Draw the radii CI, CT; then since last revolution, subtract the preceding term in the the radii of curvature are tangent to the evolute, CIB and CTD are right angles; therefore the an-, gles formed by the two radii CI, CT, and by the two radii of curvature IB, TD, are equal; hence, to {3n(n−1)+1 } in the limit, ICT and BID become similar sectors; and we shall have CI:IB::IT: BD, that is, the ra- the area described by the radius of curvature durdius of the evolute is to the radius of curvature as ing any (n.) complete revolution, or, in other the differential of the radius of curvature is to the corresponding differential of the involute, Hence, calling radius CI= a, we have words, it is the general formula for the area inclosed by any whole involute, whose number is n, and that part of the radius joining its extremities, To be more general still, we may integrate be- Expanding, x2+y+z = x2 — 2mx + m2 . tween any two limits, xh and xb; thus: puty2+x+z = y2— 2my + n2 22+x+y=22 + 2rz + r2 h3 b3 .... .... (1) (2) (3) ting xh, A =—, x=b, A! —; taking the dif- Equating the values of x in (1) and (2), also in (2) and (3), and reducing for y, we have вл 6a b3-h3 NOTE. We find an error in the first line of this part of the question (3). The word "half" was probably accidentally inserted; but it leads into no difficulty. Equating these values of y, and reducing for z, m (2n2―m) (1 — 2n) — (r2 — n2) (4mn —1) %= (2r+1) (4mn - 1) — (1 — 2n) (1—2m) Assuming now r = = 1, n=2, m = 3, we find 4 9 Again assuming r 8 16 x=1, y, z=−. 3 3 = }, n = — -1, m=-2, we find 3 Either set of answers satis fies the conditions of the problem. PROBLEM (2.) "Find such values of x and y, Suppose one end of the rope to be fastened at I. in whole numbers, as will make √(x2 + xy + y2) a less than half round the tower; the other end at Giving n and y different values at pleasure, we shall obtain corresponding values of æ. But by the conditions the results are limited to whole numbers, therefore a must be a whole number, which is evident from the second equation; hence giving to n and y proper integral values we shall obtain Restoring the values of the known quantities and corresponding values for x. [We are happy to hear from "Diophantus himself, and we admire his solutions much, but would like to have him give some discussion of the number of solutions, of various kinds, of which both whose roots are imaginary, because the signs the problems admit. When this is possible, we are all positive, and the square of the coefficient suggest whether the solution of such problems can of is less than four times the absolute term. be considered complete without it, especially since, Hence 25 is the only real root, and answers the con- in indeterminate analysis, such discussions freditions of the problem, being the length of rope quently form the most interesting part of the subrequired. ject. Problems properly embraced under the head of indeterminate analysis admit of an infinite num For the Schoolmaster. J. M. R. Solutions of the Diophantine Problems in the ber of solutions, of all kinds; but the number is November Number. often restricted when the results are required in whole numbers, or whole and positive, &c. Does PROBLEM (1.) "To find three numbers such the first problem above admit of any solutions in that, if to the square of each of them the sum of whole numbers, positive or negative? Can the valthe other two be added, the three sums shall be ues of m, n, r be assumed at pleasure? Also of squares." n and y, in the second problem? We regard the Let x, y, z represent the numbers, we then have second problem as of a more interesting character x2+y+z = 0, y2+x+2=0, 22+x+y=. than the first. We have also received similar soPut √(x2 + y + z) =x−m, √(y2+x+ z) = yn, lutions of these problems from the propounder.— ED.] Σ √(22+x+y)=z+r. Natural Science. COMMUNICATIONS for this Department should be addressed to 1. F. CADY, Warren. For the Schoolmaster. The Sense of Smell. a good example, in nature, of a function and its variable. But in human beings also this sense is at times highly developed. A beef dealer, who had contracted largely for the army, once told me, so sensitive was his nose to the qualities of beef, that on passing by eating houses in the city, he could tell whether the steak that was cooking was tender or tough, from the steam escaping from the cooking range into the street. here?" 66 It is easily seen, on a little reflection, that the faculties of observation, acting through the ageney of the eye and the ear, are susceptible of a large A blind man, riding along with a friend, not a and almost endless amount of development, giv- hundred miles from this place, was asked at a cerWhat are they doing ing us at the same time a great deal of interesting tain point on the journey, and of practical information; by means of the eye, "Well, they are doing something to a making us acquainted with the color, outline, form, house, repairing it, they are shingling a school size, direction and distance of objects, as well as house;" and he was right. Did he know that the motions and the habits of animals; thus laying they were shingling a house from the peculiar odor the foundation of drawing, painting and the study of hemlock or of white cedar, from which shingles of the natural sciences; by means of the ear, mak- are generally made? ing conversation, reading and singing possible. A year or two ago this incident was recorded in Now, what information does the sense of smell a New York paper: A physician, on entering an give us; is the faculty capable of a much fuller omnibas, at once said, "There is, in this omnibus, development than it ordinarily attains; what a case of small pox of the very worst kind." The sources of practical knowledge, now closed to us, passengers left, one excepted, a lady closely veilwould it then lay open? ed; the physician, on asking her to raise her veil, found his suspicions realized; she had been put into a public conveyance by her friends to be carried to a hospital ap town. By most persons very decided odors only are noticed; the fragrance of roses, of orange blossoms, of perfumery, or the many vile, unnamed effluvia that offend the nostrils in crowded villages and cities. No practical inferences are drawn from the presence of these odors, they being generally noticed only as pleasant or unpleasant. The chemist detects arsenic by its garlic odor, nature here having provided him with a test approximating in delicacy to the celebrated one of Reinsch. We hardly need mention, in this connection, chlorine, phosphorus slowly oxydizing, and sulphuretted hydrogen. see. In some animals this sense is developed to a degree of sensitiveness almost incredible. An old These examples, and many others might be and intelligent hunter once remarked to me, that brought up, show us how exceedingly delicate the even when a fox-track is three days old, a hound sense of smell at times becomes, in some men, needs to smell of but few of the impressions left and make it probable, that, with care, it might in the snow, in order to get the scent and to follow be more fully developed in others than it now is. it aright. The hound scents the fox by means of But what good would result from this. Let us the little particles left in each of his tracks by the feet of the fugitive, and how small these particles Each simple and each compound, if in the least must be, when the fox runs ten miles, as he often volatile, has its peculiar odor, sometimes strong, does, before the hounds, his feet, at the end of the sometimes faint. Of these substances, some are heat, being apparently in a natural state! How injurious to health, some not; some are in snch these particles also, in three days, must waste places that their existence cannot be detected, away, from decay, exhalations and from sinking though the particles which escape from them in every direction, betray, to one not heedless, But the dog knows which way the fox went, their place of concealment. In this way one should how does he know this? If 1-2-3-4, are be able to know when sinks, drains, sewers, vaults, the foot-prints in the order in which they are made, stables or decaying animal and vegetable substanthe dog perceives such a difference between 1 and ces are poisoning the air in the neighborhood, and 2, 2 and 3, 3 and 4, that he knows 2 was made af- the nuisance should be abated. The teacher should ter 1, 3 after 2, &c.; the scent then in 2 must be be able to tell whether or not the school-room be into the snow! -- stronger than in 1, in 3 than in 2, &c. These properly ventilated, or if there is coal gas escapscents differ in this only, that 1 has faded out more ing from the stove. than 2, 2 more 3, &c., and 1 is as much fainter An enormous amount of drugs is used in this than 2, as I would fade out in the time intervening country, most of which are largely adulterated. between the fox in his flight making 1 and making It is well to have an agent, appointed by govern 2, yet this difference, small as it is, the dog detects. ment, to inspect those that are imported, but a It seems to me that the scent of a fox track offers better safeguard against imposition and loss, in slope. many cases, would be furnished by the customer's had oozed out from the walls and from the slippery noticing the odor of the article offered for sale. Smell of the compound sold by apothecaries as The main shaft, extending in a direct line to the "Madeira" or "Port," when your physician pre- bottom, is intersected by several parallel, crooked scribes some wine in a case of sickness. Do you passages running northerly and southerly. Its anthink that grapes could ever yield so digusting angle of inclination may be somewhat greater than odor? How does the fragrance exhaling from this 45 degrees, so that the bottom is about 350 feet, chest marked " English Breakfast Tea" remind perpendicularly, from the surface. Two rail-ways you of meadow hay and of swamp herbs! How serve for the passage of heavy, Iron box-cars, prostrong the odor of fusel oil from this mixture, the pelled by means of ropes attached to an engine latest patented, and before you purchase any more stationed at the entrance of the mine. The coal of a "Burning Fluid" warranted "not explosive," obtained here is not of the best quality. It is disnotice the strong smell of turpentine exhaling from posed in strata perhaps eight feet in thickness, beit, and then purchase it at your children's peril. tween alternate layers of slate rock. It appears, then, in answer to the third part of An interesting geologic phenomenon are those our inquiry, that the sense of smell, like the sense boulders found scattered over the meadows and of feeling residing throughout the surface of the collected in vast fields on the sides of hills in the body, is given us as a friendly monitor, to warn us towns of Exeter and Richmond. After passing of the neighborhood of substances injurious to the summit of Pine Hill, the traveller, turning to health, which the eye either does not or cannot the left across lots, just below the toll-gate on the perceive; and it is as unwise to be heedless of the New London pike, will find a field of large stones, monitions of the one sense as of the other. There at first scattered, then closely collected, and tois folly in allowing a manuf ctory for making sul-wards the middle of the field piled up to the depth phuric acid to spread its deaaly fumes over a town, of ten or fifteen feet. These are all rounded bould as well as in permitting malicious boys to scatter ers, of granite or sienite, as far as I have ascertainthe acid itself over the passers-by. A current of ed unlike any of the stationary rock in the vicinity. air in passing over gas-works is thereby poisoned, They appear to be mostly composed of feldspar of and no person who values his health will live with- a red tint, quartz and hornblende, and contain in a hundred rods of such works, or within a square little or no mica. of a stable. And this, not on account of the of fensiveness merely of such localities, but because of their unhealthiness. We also see that this sense may serve to protect us from imposition in making purchases, so as not to buy as genuine, articles adulterated, unhealthy and worthless. And, Mr. Editor, I hope to see the day when we shall have become so sensitive as not to endure the air in rooms heated by an "air-tight" stove, not to tolerate in our parlors one whose breath betrays that he "drinks" or smokes, one who does not bathe frequently, or who puts any of those vile compounds called "blacking" on shoes, or "hair oil" on the head." For the Schoolmaster. J. K. Further north, in the town of Johnston, the gra nite is of very different appearance, being lightcolored and close-grained. The Arcade pillars in Providence are hewn from Johnston granite. At Valley Falls, Smithfield, we again meet coal. In this case, the dip is rather towards the north or north-east, at about the same angle as that of the Portsmouth mine. The proprietor grinds the debris from the mine into a powder suitable for "facing for founders' use. This shaft does not appear to I have been be extensively worked for its coal. successful in obtaining from the slates thrown out of the mine when it was worked, some beautiful petrifactions. Among these I have retained a leaf of the annularia, a portion of petrified wood, several ferns, a few delicate leaves, and also a portion of the large rough leaf of a gigantic plant. Dr. Jackson's geology of Rhode Island guided THE island of Rhode Island is probably the re- me among the lime-quarries of Smithfield. My sult of an upheaval of the coal and slate strata field book, dated February 1, 1859, contained copi from both sides. On its east shore, the strata dip ous notes, but as I followed in the track of Dr. towards the centre, and from its west side in the Jackson, I did not preserve them. I did not find same direction. Unstratified rocks are common the rock on Lonisquisset turnpike indicated by in the western and southern part of the State. him as a whetstone rock, although I inquired after The party that explored the coal-mine on Rhode it in the neighborhood. It is not probably worked Island, on a pleasant summer day, at the time of at present. My route home led me by a large manthe meeting of the State Institute at Portsmouth ufacturing establishment, where my bag, loaded there were seven of us will remember how wea- with rocks, attracted the attention of a vigilant rily every one found the ascent of six hundred feet watchman, who, supposing me to be a pedler, hintout of the mine to be, and with what readiness we ed at sending me back by the way I came. However, after a short parley, he suffered me to pass. sought the hot-water tank that we might wash My design was misconstrued. Such is the fate of away some of the black mud adhering to us that scientific enthusiasm! J. W. O. |