EXERCISE TO PROPOSITION XLIV. equal to the complement K D, to each adding the parallelogram AEKH, therefore the whole AG is equal to the whole AF (Ax. 2); wherefore, the triangle BAH is equal to the triangle EAD (Ax. 7). From each of these equals, taking the triangle EAH, the remainder the triangle BEH is equal to the remainder the triangle E HD (Ax. 3); but these triangles are on the same base, EH, therefore the straight lines EH and BD are parallel (1. 37). In the same way, it may be shown that GF is parallel to BD; therefore EH is parallel to G F (I. 30); and EH, BD, and G F are parallel to one another. Q. E. D. EXERCISE II. TO PROPOSITION XLIII. If about the diagonal of a parallelogram, any number of paral lelograms be placed, whether the extremities of their diagonals coincide in the same point or not, the remaining complementary rectilineal figures on each side of the diagonal of the parallelogram, are equal. In fig. 47, let A B CD be a parallelogram, and ▲ c its diagonal; and let EG, LI, and NP, be any number of parallelograms placed about the diagonal in any manner whatever; the complement EF HLKM NOCD is equal to the complement Fig. 47. To a given straight line to apply a triangle equal to a given parallelogram, and having an angle equal to a given rectilineal angle. By the exercise to Prop. xxxvii., describe a triangle equal to the given parallelogram; and by Prop. xlii., to the given straight line, apply a parallelogram equal to this triangle and having one of its angles equal to the given rectilineal angle; then by Prop. xxxvii., describe a trianglequal to this parallelogram, so that the angle in it which is equal to the given rectilineal angle, and the side in it which is equal to the given straight line, may be respectively an angle and a side of the triangle. Then by construction, and Axiom I., the thing required is done. ‡ EXERCISE TO PROPOSITION XLV. If in the sides of a square, points be taken at equal distances from its four angular points in succession, the straight lines which join these points, will form a square. This square will be less than the original square in proportion to the distance of the four assumed poin's from its angular points until that distance be equal to half the square, when the inscribed square (Def. I., Book IV.) will be a minimum, that is, the least possible square which can be thus inscribed. In fig. 48, let A B C D be a square, and E, F, G, and н be four Fig. 48. points taken at equal distances from its angular points A, B, C, and D, in succession; then first, the straight lines E F, FG, GH, and HE, which join these points, form a square E F G H. Because AE is equal to BF, and A B to B C (Hyp.), therefore EB is equal to FC (Ax. 3). In the same manner, it may be shown that AH is equal to E B. Because in the two triangles EAH and F BE, the two sides E A and AH are equal to the two sides F B and B E each to each, and the angle E A H is equal to the angle FB E, each being a right angle, therefore (I. 4) the side EH is equal to the side EF, and the triangle E AH to the triangle FB E. In the same way it may be shown that the side FG is equal to the side EF, and the side G H to the side HE; therefore the four sides E F, FG, GH, and H E are all equal to each other (Ax. 1), and the figure is equilateral. Because the triangle EA H is equal to the triangle FB E, the angle A EH is equal to the angle B FE, and the angle EHA equal to the angle FEB (I. 4). Because the exterior angle HEB of the triangle EA H, is equal to the two interior angles EA H and AHE (I. 32), and the angle FEB is equal to the angle E H A, therefore the remaining angle F BH is equal to the remaining angle EAH; but EAH is a right angle, therefore FEH is a right angle. In the same way it may be shown that the angles EFG, PGH, and GHE are right angles; therefore the figure EFGH is equiangular and right-angled; and it was proved to be equilateral and four-sided; therefore it is a square, Next, let 1, K, L, and м, be the middle points of the four sides of the square ABCD; it may be shown in the same manner, that the straight lines IK, KL, LM, and MI, which join is the least square which can be thus inscribed in the square these points, form a square IK LM. Also, the square IK L M ABCD. Let EFGH be any other square. Join MK and HF, and let these straight lines intersect each other in o. Solved by QUINTIN PRINGLE (Glasgow); E. J. BREMNER (Carlisle); WARIN (Dereham); and J, H. EASTWOOD (Middleton). T Because MD is equal to в K and D H to BF (Hyp.), therefore MH is equal to FK (Ax. 3). Because in the two triangles MO H and FOK, the two angles o M H and о H м are equal to the two angles OK F and OFK (I. 46 and 29), therefore the triangle OMH is equal to the triangle o KF, the side мo to the side o K, and the side Ho to the side or (I. 26). Because MD is equal and parallel to KC, therefore м K is equal and parallel to DC (I. 33) and O M D is a right angle (I. 46 Cor.). Because in the triangle oм II the angle oм н is a right angle; therefore it is greater than the angle o H M (I. 32), and the side o H is greater than the side o м (I. 19), and the whole FH than the whole MK; for it was proved that they bisect each other in o. Because FH is the diagonal of the square EFG H, and м K the diagonal of the square IKLM, therefore the square E F G H is greater than the square I KLM (I. 43). In the same manner it may be proved, that any other square whose four points are not taken at the distance of half the side of the square A B C D, from its angular points A, B, C, D, is greater than the square IKLM. Therefore the square I KLM is a minimum, that is, it is the least square which can be thus inscribed in the square ABC D. Therefore, if in the sides of a square, etc. Q. E. D. ANSWERS TO CORRESPONDENTS. P. ALEXANDER (Glasgow): The Latin essay is very well for a first attempt; persevere and you will succeed.-J. WUDROW (Kilmarnock): We don't know of any cheap books on Mechanical drawing. The Society of Arts, we believe, supplies Mechanics' Institutions with the cheap box of Instruments.-T. B. CRABTREE (London): "A little learning is NOT a dangerous thing." Learning, however little, is always useful, and to some essential; but then it must be learning, and not pretence to learning. Let a man really know ever so little, if he but knows something, he will be sure to know more very soon. There is no danger in learning, but there is great danger in the want of it. For the ordinary duties of life, the branches of learning are Reading, Writing, and Arithmetic, all of which we have given in the P. E., and a great deal more. Moreover, we say that if a young man will only study the first four volumes of the EDUCATOR, and we don't mean the Foreign Languages, unless he likes, he will do more good in society than if he had been seven years at school in any town or village in the kingdom. Your name shall be forwarded to the Petition; but persevere in your studies, and remember that calling names is no argument. 1 Peter, chap. 1. verse 20, BETH (York): Human language cannot be employed without imperfection in speaking of God, especially of his attributes. Thus, God is good, and he being absolutely perfect in goodness, we cannot say that God is better at one time than at another; but we can say that he is better than men or angels, or any created being; and we can say that he is the best of all beings. Again, although when we say the cup is full, the cup cannot contain more than full; yet it can contain more than half-full or three-quarters full; and in the latter state it is fuller than in the former. Incumbent (Incumbens), vicar (vicarius), and rector are Latin words; see Cassell's Latin Dictionary. -J.CAUSTON (Southgate): Thanks.-H. JUPP (Leith); W. ONN (Nottingham); W. P. BUSK (Chadderton); G. ARCHBOLD (St. Peter's); GEO. SMITH (Manchester); and many others, have solved the lady's age, and the pinetree questions, as follows: Let be the lady's age, then 2 X 3 X 2 X 3x = 4 or 9 × 7 √ {8 (3 { # (3x) }) } = whence 28. Or, without Algebra. If first you square the number four, Take now seven-halves, and then once more, Of eighty-four, the product, take One-third, and the result will make Just twenty-eight, the lady's age. Please put this in your Answer-page.-P. V. = 16; is the 9600; Let be the length of the stump of the pine-tree; then 100 length of the broken piece; whence, by the question and Euclid I. 47, we have +400 1000 200x + x; and by transposition, etc. 200 therefore x = 48, the stump; and 100 = 52, the broken piece. MATHILDE (Brompton): Her style of poetry is pretty fair; but we don't admire the subject of the piece she has sent. We are humble advocates of peace, and we could not feel justified in saying, "Fear not, O Mussulman, To return the blow." A. ROBINSON (Bradford): His name is forwarded.-T. T. (Edinburgh): His suggestions are very good, but we cannot afford time for the research. -R. D. DAVIS (Merthyr Tydvil): Very well; go on and prosper.-JOHN HALL: He will not.-R. BOURNE (Darlington); LABORE VINCO (Soho); G. WILD (Dalton); P. ALEXANDER (Glasgow); J. HORSFIELD (Wyke); W. C.; BLANDUS; T. W. PARDOE; J. HONE (Bloxwich); C. H. KENNION (Farnworth); W. F. HILLS (Mildenhall); C. THOMAS (St. Austell); J. JONES (Woolwich); J. POGSON (Mossley); MARIA M. LEATH (Otley): Their solutions of the Pine Query are all right.-R. PARKINSON (Everton); P. ALEXANDER (Glasgow); J. HONE (Bloxwich); J. H. STAUNTON (Chel sea); J. WILKINSON (Guildford): CONST. SUBSCRIBER (Langham): W. F. HILLS (Mildenhall); C. THOMAS (St. Austel); M. M. LEATH (Otley): Their solutions of the Lady's age are all right.-J. WILKINSON (Guildford); H. DAVIS (Maida Hill): Their solutions of the Marriage Query are all right.-J. HONE (Bloxwich); W. C.; R. PARKINSON (Everton); W. F. HILLS (Mildenhall); J. II. STAUNTON (Chelsea); J. POGSON (Quick of the Two Tower Query are all right.-T. W. PARDOE proposes the folView); H. DAVIS (Maida Hill); W. ONN (Nottingham): Their solutions lowing query, to which we have made additions: "The summit of a tower was found One hundred feet above the ground; Against the wall a ladder stood, Which did that very height make good. A rogue this ladder did make slide, Exactly ten feet from the side, Just at the bottom; while the top Its sliding notion then did stop. How far, below the summit, fell The ladder's top? Pray students teli. And if the sliding did proceed As it began (to this give heed), While one the ladder did ascend, And at the moment did intend To reach the ground, when at the top, And when the sliding then should stop. How far would he above the ground, When at the highest point, be found? And what would be the curve his heel Would thus describe? Pray now reveal. J. HONE (Bloxwich); C. THOMAS (St. Austell); J. POGSON (Mossley) Their solutions of the Star Query are all right. LITERARY NOTICES. 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Parts 1. and II.-Price 28. each, in paper covers, or 2s. 6d. in cloth. Two Parts bound together, price 4s. 6d. A Key to the above Lessons is in the Press. CASSELL'S ECLECTIC GERMAN READER: containing choice Selections from the best German Authors, in Prose and Verse. Price 2s. paper covers, or 2s. 6d. cloth. CASSELL'S LESSONS IN GERMAN PRONUNCIATION: consisting of easy Extracts from German Writers. Price 18. paper covers, or 1s. 6d. eloth. CASSELL'S LATIN GRAMMAR. By Professors E. A. ANDREWS and S. STODDARD. Revised and Corrected. Price 3s. 6d. in cloth boards. CASSELL'S SHILLING EDITION OF FIRST LESSONS IN LATIN. By Professors E. A. ANDREWS and S. STODDARD. Revised and Corrected. Price Is. paper covers, or 1s. 6d. neat cloth. ON PHYSICS, OR NATURAL PHILOSOPHY. No. XLV. (Continued from page 274.) [OPTICS. TRANSMISSION, VELOCITY, AND INTENSITY OF Nature of Light.-The agent which produces in us, by its action on the retina of the eye, the phenomenon of vision, is called light. That part of physics which teaches the properties of light is denominated optics. In order to explain the origin of light, two hypotheses have been adopted, as in the case of heat; viz. that of emission, and that of undulations. In the former, adopted by Newton, it is considered that luminous bodies emit in all directions, under the form of particles of extreme tenuity, an imponderable substance which is propagated in straight lines with a most prodigious velocity. These particles entering the eye, act upon the retina, and produce the sensation which constitutes vision. In the hypothesis of undulations, adopted by Grimaldi, Descartes, Huyghens, Young, Malus, and Fresnel, it is considered that the particles of luminous bodies are animated by a vibratory motion, infinitely rapid, which is communicated to an eminently subtle and elastic fluid which is diffused throughout the universe, and which is called ether; and that a disturbance at any point easily to pass through them, and which enable us to see objects through them, are called diaphanous or transparent bodies, such as water, the gases, and polished glass; those which enable us still to perceive light through them, but not distinctly to recognise the forms of objects, are called translucid or translucent, such as ground-glass and oiled paper; and those through which no transmission of light takes place are called opaque, such as wood and metal. Still there are no bodies completely opaque; all are more or less translucid when they are reduced to plates or leaves sufficiently thin. The track or line which light takes in its propagation is called a luminous ray; and a collection of such rays emitted from the same source is called a pencil of luminous rays. A luminous pencil is called parallel, when it is composed of parallel rays: divergent, when it is composed of rays which diverge or separate from each other; and convergent, when the rays all meet in the same point. Every luminous body emits from all points and in all directions, rectilineal divergent rays. The space full or empty of diaphanous matter in which the phenomena of light are produced, is called a medium (plural, media). Air, water, and glass are media in which light is propagated. A medium is called homogeneous (that is, of the same kind) when in all its parts its chemical composition and density are the same. In every homogeneous medium, light is propagated in straight lines. Hence, if we interpose an opaque body between the eye and a luminous body, on the straight line which joins them, the light is intercepted. It is observed also that the light which penetrates into a dark room, through Fig. 243. M P a small aperture, traces in the air a luminous straight line, which becomes visible by illuminating the slight particles which are held in suspension in the atmosphere. Yet light changes its direction when it meets an obstacle into which it cannot penetrate, or when it passes from one medium into another; these phenomena will be explained under the heads of reflection and refraction. whatever of this ether is communicated in all directions under the form of luminous spherical waves, in the same manner as sound is propagated in the air by sonorous waves. Yet it is considered that the vibrations of the ether are produced, not so perpendicularly to the surface of the wave as in the propagation of sound, but along this surface, that is, perpendicularly to the direction which light takes in propagating itself; and this is expressed by saying that the vibrations are trans- Umbra, Penumbra, etc.-The umbra or shadow of a body is the versal. We may form an idea of these vibrations by shaking part of space which it prevents the light from entering. When a string when held by one of its ends; the motion is propa- the extent and the form of the shadow projected by a body gated in a winding form to the other end; here, therefore, are under consideration, two cases may arise; viz. that in the propagation takes place in the direction of the cord, but the which the luminous source is a single point; and that in which vibrations are made crosswise or transversally. By the sys- it is a body of some size or extent. In the first case, let 8, tem of undulations, Fresnel has been enabled to give a com- fig. 243, be the luminous point, and м the body which proplete explanation of many curious phenomena of light, such jects the shadow, and which we shall suppose to be spherical. as those of diffraction and of coloured rings, which cannot be If we conceive that an indefinite straight line, s a, moves explained on the system of emission. The theory of undula-round the sphere M, always remaining a tangent to it, and contions, indeed, has been generally received since the publication of Fresnel's researches. Definitions.-Bodies which, like the sun or a candle, emit light, are called luminous bodies; those which permit the light stantly passing through the point s, this straight line will generate a conical surface, which, on the side of the sphere opposite to that where the point s is placed, separates the portion of space which contains the shadow from that which Fig. 244. is illuminated. In this case, by placing a screen, PQ, at right angles to the straight line from s passing through the centre of the sphere, in the place occupied by the shadow, but sufficiently large also to intercept the luminous rays, the transition from the shadow to the illuminated part will be distinctly marked; but this does not take place in ordinary cases, where the luminous bodies have always a certain magnitude. In order to simplify the explanation of this matter, let us suppose that the luminous and the illuminated bodies are two spheres, s L and м N, fig. 244. If we conceive that an indefinite straight line, A G, moves round tangentially in contact with these spheres, and constantly passing through the point A in the straight line which joins their centres, it will generate a conical surface which has for its vertex this point A, and which limits on the other side of the sphere M N, a space MGHN completely deprived of light. If a second straight line, LD, cutting the straight line which joins their centres in B, moves round also tangentially to the two spheres, so as to generate a new conical surface, B D C, it will be seen by the inspection of the figure, that all the space exterior to this surface is completely illuminated; but that the part comprised between the two conical surfaces is neither completely deprived of light, nor completely illuminated. Hence, if we place a screen, Pa, at right angles to the straight line passing through the centres of the spheres, and intercepting the dark, illuminated, and semi-illuminated spaces, the portion cod of this screen will be completely in the shadow; and the annular portion ab will receive light from certain dresses, employ the artificial effects of the double reflection which light affords. Passage of Light through small Apertures.-When we receive on a white screen, the luminous rays which enter a dark room by a small aperture, we obtain from exterior objects, images which present the following phenomena: 1st, the objects are inverted; 2nd, their form, which is always that of the exterior objects, is independent of the form of the aperture. The inversion of the images arises from the circumstance that the luminous rays which proceed from the exterior objects and enter the dark room, cross each other in passing through the aperture, as shown in fig. 245. As they continue to be propagated in straight lines, the rays proceeding from the highest points of an object fall upon the screen at the lowest points; and conversely, those proceeding from the lowest points fall upon the screen at the highest points; thus the image is naturally inverted. When we treat of the camera obscura (that is, the dark room), we shall show how the brilliancy and the distinctness of the images are increased by means of convergent lenses, and how they can be exhibited in an erect and not in an inverted position. In order to show that the form of the image is independent of the form of the aperture, when the latter is sufficiently small, and the screen sufficiently large, let there be a triangular opening at o, fig. 246, made in the window-shutter of a dark room, and let there be a screen, a b, on which the image of a candle-flame, A B, placed exteriorly to it, is permitted to fall. From every point of the flame there penetrates into the dark room, a pencil of divergent points of the luminous body, but not from all. This portion of the screen is, therefore, more illuminated than the shadow, properly so called, but it is less enlightened than the rest of the screen; on this account, it is called the penumbra, that is, nearly a shadow. These shadows as now explained may be denominated Geometrical shadows; but the Physical shadows, that is, those which we actually observe in nature (physics), are not so distinctly limited and marked out, as it were, by lines of demarcation. In the case of actual observation, we find that a certain quantity of light passes into the umbra or shadow, and vice versa. This phenomenon, which will be explained in the sequel, is called Diffraction. When an opaque body intercepts light by one of its sides, the opposite side is never completely dark; it is always more or less enlightened by the light which is reflected from the surrounding bodies; this is the effect of the reverberation of light, which may be denominated double reflection. As the light reflected by a coloured body partakes, in general, of the colour of these bodies, it follows that the double reflection assumes the tint of the surrounding bodies. Artists in their pictures, scenepainters in the choice of their draperies, and ladies in their rays which form on the screen a triangular image, similar to the opening, as shown in the figure. Now, it is the assemblage of all these partial images which produces a complete image of the same form as the luminous object. Indeed, if we conceive that an indefinite straight line moves round the opening in the window-shutter, supposed to be very small, under the condition that the straight line always remains tangential to the luminous object, A B, we can suppose that in its motion the straight line describes two cones, having for their common vertex the aperture in the dark room, and for the one base the luminous body, and for the other the illuminated part of the screen, that is, the image; hence, if the screen be perpendicular to the straight line which joins the centre of the opening with the centre of the luminous body, the image will be similar to that body; but if the screen be oblique, the image will be extended in the direction of the obliquity. We observe this phenomenon, for instance, in the shadow projected by the foliage of trees; the luminous pencils which pass through the leaves give images of the sun which are round or elliptical, according as the ground on which they are projected is perpendicular or oblique to the solar rays; and this is the case, what ever may be the form of the spaces between the leaves through which the light passes. The Velocity of Light.-Light is propagated with such a velocity that we cannot, at the surface of the earth, establish any perceptible interval, whatever may be the distance, between the instant when a luminous phenomenon takes place, and the instant when it is perceived by the eye. It was by means of astronomical observations that this velocity was first determined. This discovery is due to Romer, a Swedish astronomer, who was the first, in 1678, to deduce the velocity of light from the observation of the eclipses of the satellites of Jupiter. We know that Jupiter is attended by four satellites, which revolve round him with rapidity, in the same manner that the moon revolves round the earth. One of these satellites, fig. 247, undergoes its immersions, that is, its entrances into the shadow projected by Jupiter, at equal intervals of time, viz. 42h. 28m. 35s. While the earth is in the part ab of its orbit, that is, sensibly at the same distance from Jupiter, it is observed that the intervals between two consecutive immersions remain constant, or always the same; but in proportion as it departs from this position, in its revolution round the sun s, the interval between these two immersions increases; and when, at the end of six months, the earth has passed from the position r to the position r', it is observed that a retardation amounting to 16m. and 26s. has taken place between the instant when the phenomenon appears to happen, and that instant when, according to calculation, it has in reality actually taken place. Now, when the earth was in the earth from the sun, or about 190 millions of miles; from this, it is easy to calculate the velocity of light per second, namely, 193,000 miles per second; or, in round numbers, nearly two hundred thousand miles. In this observation, the motion of Jupiter is neglected; for this planet takes nearly twelve years to describe its orbit round the sun; and consequently it describes only one-twenty-fourth part of it in six months; hence, in this approximate calculation, it may be considered as immoveable. The stars nearest the earth are at least 206,265 times more distant than the sun. The light which we receive from them, therefore, takes more than 3 years in order to reach us. The stars, however, which are only visible by means of a telescope, are at such a distance from the earth, that it would take thousands of years to elapse before their light could reach our planetary system. These stars might therefore be extinct for ages, while we should still continue to see them and study their motions. Foucault's Apparatus.-Notwithstanding the prodigious velocity of light, M. Foucault has invented a method of determining it experimentally by means of an ingenious apparatus, constructed on the principle of the revolving mirror invented by Mr. Wheatstone, to measure the velocity of electricity. Fig. 248 represents, on a horizontal plane, the principal arrangements of M. Foucault's apparatus. The window-shutter K, of a dark room has a square aperture in it, behind which is stretched vertically a fine platinum wire o. A pencil of solar light, reflected exteriorly from a mirror, enters the dark room by the square aperture, meets the platinum wire, and thence is position T, the solar light reflected by the satellite E had, in order to reach the earth, to describe the distance ET; while in the second position T' the light had to describe the distance E'T', which exceeds the former by the quantity TT'; as the rays ET and ET' may be considered as parallel. It is evident, therefore, that light must describe the diameter TT' of the earth's orbit, in order to reach the second position, in 16m. 26s. of time; but this distance is twice the distance of the made to pass through the focus L of an achromatic lens placed at a distance from the wire less than double the distance of the principal focus. The image of the platinum wire tends then to form itself on the axis of the lens under dimensions more or less enlarged. But the pencil of light, after having passed through the lens, meets a plane mirror m, revolving with a great velocity, from which it is reflected and proceeds to form in space an image of the platinum wire, which changes |