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(m, k,) (l, n,) vertical, and the machinery in action; it will be seen, by examination, that motion would be communicated to the vessel, but that it would be simply vertical, a mere up-and-down movement, and that the deck would always be parallel to the line in which it lay at starting. If we add the lever (r), centring it midway between the centre-pins of the shafts (0, p.), a very small, but scarcely a perceptible variation, would be produced; but if now we place its centre-pin (s) nearer to the centre-pin (p) of one of the shafts, than to that (o) of the other, we shall have the motions of the centre-pins so controlled by the radius (s t), that they move, both ascending and descending, with different and differing velocities; so that the stem and the stern of the ship will rarely remain for two successive instants in the same level plane.

In the following diagram (fig. 2), are shown the positions of the deck, which correspond to four successive and simultaneous positions of the cranks. The arrrows indicate

Fig. 2. the direction in which the cranks turn round.

When the cranks stand at o A, the deck will be in the position 8 T; as the cranks move to the position 0 B, s will ascend to u, and T descend to V, and the deck will be at u v; during the change of the cranks to oc, U will descend to W, v to x, and the deck will attain wx; let the cranks go on to o D, w will now ascend to y only, but x to z, y z becoming the position of the deck; as the cranks go on to the starting positions o a, y will ascend to s, and z descend to T, the deck will arrive at st, the position whence it set out. It may therefore be seen, that in each interval of time, the motions of the stem and of the stern are different, one of them being always greater than the other, and that at two points in the course, the one which was the greater becomes the lesser, and vice versa. It is owing to the ingenious introduction of the lever (r) into its peculiar position, with regard to the shaft centre-pins (s,p,) that this play of changes takes place, and the pitching of a ship in a brisk gale and high-running sea, is so beautifully imitated. By the weight (u) this pitching can be made quicker or slower, at pleasure.

The invention is French, and patented. The names of T. C. CAILLY and EUDE, are stamped upon the machinery-case.

Magnetic NEĒDLE; Monday, 29th February. THE council of the Society directed that a very large magnetic needle should be immediately constructed, and fitted up in the most careful manner, by Mr. Saxton; and be deposited in the Gallery, for the purpose of effectively showing the variation, oscillation, &c., of the magnetic needle, at London.

REVIEW.

I. Optical Investigations. 1. Caustics. 2. Optical Images. By the Rev. G.

H. S. JOHNSON, M.A., Tutor of Queen's College, Oxford. Published by

Talboys, for the Mathematical Society. We congratulate the University of Oxford on the institution of a Mathematical Society. This body, though we are informed it is extremely limited in point of numbers, is yet, we conceive, likely to be of great importance in furthering and encouraging mathematical studies, so little generally pursued in that university, yet, as we have now abundant evidence, carried to so high a point by some few distinguished individual members. If it do nothing more than act as a printing-machine for memoirs like those now before us, it will do much; but we trust that its meetings may yet be productive of more extensive good. This society has as yet published only the two papers above-named, which, though printed as separate tracts, may in fact be regarded, in connexion with a previous publication of a similar kind by the same author (relating chiefly to observations), as a series of investigations on some of the most important topics in what is distinctively called mathematical optics.

With regard to the first, the subject of caustics is one which we cannot say much to illustrate, in the compass of such a notice as the present. A caustic is the luminous line or curve formed by the intersections of rays of light after reflection, at curved surfaces, or after refraction through media bounded by such surfaces. A familiar example is seen in the inside of a tea-cup, when the rays of a candle shine into it. On the surface of the liquid there will be seen a curved line of light, whose form, relatively to the circular outline of the сир, is represented in the annexed sketch.

Such curves, formed under different circumstances, are an elegant subject for geometrical investigation; and have been treated by Mr. Johnson, in the tract before us, in a very original and beautiful manner, accompanied by some illustrative examples, which we believe

are new.

The subject of the second tract is one referring to matters more familiar to every one who has seen optical experiments, and of more general interest; and on which it may be worth while to say a few words, to remove an ambiguity into which some writers fall. An optical image may be defined to be the formation of a distinct point of light for every point in the object whence a ray of light originates, and in the same relative position; so that these points of light, collectively, (if their positions were marked,) would give rise to a picture or resemblance of the object, either exactly similar, or distorted in some of its proportions, according to circumstances. VOL. I.

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Such a formation of corresponding luminous points may take place in several different ways; and the position of those points, and consequent depicting of the image, may be produced in several ways; but the essence of an optical image is their actual formation by whatever means.

Thus, in the simplest case of the camera-obscura, the hole in the shutter and picture of external objects on the wall: from every part of the object (suppose the flame of a candle), a body of rays reach the hole. The minute aperture only allows a single narrow ray to pass; this reaches the wall or skreen, and, being thus limited and defined, and of sufficient intensity (since it is contrasted with the darkness of the room), will give (at whatever part of its course it may be stopped) a luminous point: all the other rays do the same in their respective directions; hence, at any distance, an image is painted on a screen.

Next, let us take the case of a plane reflector. From the flame of a candle, as before, rays fall on every part of the mirror, and are reflected: but on no wall or screen is an image of the candle produced: The image is often said to be formed behind the mirror. Even suppose the silvering removed, a screen placed behind will have no image depicted on it. When, then, is it formed? or is there any image at all? We reply there is none in the sense commonly adopted; but there is one formed in the eye (as a camera-obscura,) by the precise process of the last paragraph (modified only by the introduction of a lens at the aperture). It is matter of easy investigation, however, to show that these rays which fall upon the eye, come in the same directions as they would have if they proceeded from a real object, situated at the same distance behind the mirror as the eye is before it.

The same thing precisely may be said of convex reflectors. The eye is deceived into the belief that an image exists behind the mirror; and, from the course of the rays, we may speak mathematically of an image existing there, but the whole is a geometrical fiction.

With a concave mirror it will be said the case is different: here we can really exhibit the image on a screen: it is actually formed in the air. The eye, again, may be so placed as to see it. But these are two totally distinct cases. They are each separately simple cases of optical images; but they are not related to each other: the image, as formed in the eye, is not an image relatively to the mirror. The rays come from the mirror as from an object, and they form an image in the eye as a camera-obscura. In the other case, the image is really one relative to the mirror, and is formed (agreeably to our definition) by the small definite pencils of rays which, from each point in the object, come by a separate course to give each a luminous point, corresponding to the point of its origin, and thus, collectively, an image.

This is, again, the case with lenses. The essential point, as before, is here the formation of a separate defined focal point of light for each luminous point in the object. When these are formed collectively, there is an image. In convex lenses it can be depicted on a screen; it can also be perceived by the eye. But the two cases are again essentially distinct; the one is an image belonging to the lens; the other to the camera-obscura of the eye.

Now the image formed in the eye, by the rays proceeding from a refracting surface, or through a reflecting medium, as if from an objeci, will be modified in a variety of ways, according to the course which those rays take; and the object of the mathematical inquirer is to investigate what directions will be given to rays, supposed, in the first instance, to come from a luminous point, and then to be reflected, or refracted, according to the known laws of reflection or refraction, at surfaces either plane or of any given geometrical species of curvature. According to these conditions they will seem to come, as it were, from an object of very different shape, and the determination of this constitutes what is termed the mathematical investigation of optical images. To this subject Mr. Johnson has directed his powerful mathematical abilities, and has treated it in a very simple and elegant, if not wholly original manner. It would, of course, be entirely unsuitable to our pages to enter upon details; but we will mention one result at which he arrives, which is somewhat singular. A straight line, or stick, wholly immersed in water, always appears to the eye sensibly straight, though its apparent position is changed. Mr. Johnson was conducted, by his mathematical analysis, to a certain algebraic formula, which ought to express the nature of the image in this case; and instead of a straight line, (which would, algebraically, be expressed by a simple equation, he found a curve of a high and complex, order. When, however, he proceeded to inquire more closely into the nature of this curve, he found that one part or branch of it only, was actually concerned in the problem: and that branch was found to have the remarkable property of taking a form so slightly curved that to the eye it would be sensibly confounded with a straight line: thus evincing the singular accordance of geometry with nature in the midst of apparent discordance.

II. Perspective Rectified; or, The Principles and Application Demonstrated.

With a New Method of Producing Correct Perspective Drawings without the Use of Vanishing Points. By ARTHUR PARSEY, Professor of Miniature Painting and Perspective. 4to., 16 Plates. London, Longman

and Co. We perfectly agree with Mr. Parsey in his opinions, both as to the advantages of a knowledge of perspective, and the facility of acquiring that knowledge by any one inclined to take the pains. Mr. Parsey has obviously thought much and justly on the principles of art: many of his observations are original and important. We are, therefore, surprised, that with his qualifications, he should have failed to see that Linear Perspective is strictly a branch of geometry, which can neither be learnt nor taught, but by a rigid adherence to mathematical deduction and demonstration. The fact, as we conjecture, is that the author has studied the subject more with the feeling of an artist than with that of a geometrician, for his train of reasoning and language are deficient in that precision and accuracy so essential to all mathematical investigations.

There is no question but that the apparent forms of objects, as impressed on the retina, are modified by the construction of the organs of · vision; it is these forms that the artist endeavours to transfer to his canvas when he draws by eye, as it is termed, and his power of doing this

with judgment and facility will doubtlessly be increased by a general acquaintance with the rules of linear perspective: but this last-named art waves all optical considerations, and only professes to furnish the means of delineating on a surface, the contours of figures, bounded by simple geometrical lines and surfaces, as they would be seen by an eye considered as a geometrical point: hence it is, that all delineations obtained on these principles are, in fact, incorrect, when viewed from any other than the precise point from which the outlines were deduced, and which is taken to represent the eye of the spectator; the geometrical draughtsman, however, aware of this, shows his judgment by selecting his point of view, so that no obvious distortion may be apparent in his outline, when viewed from any point indifferently as drawings usually are.

It is true that the parallel lines of the top and bottom of a long wall would always be projected on a plane into straight lines, but they are projected into curves on the concave retina of a person standing opposite the middle of the length of such a wall; and the lines in question do, to him, seem to approach each otheon either side of him, and yet present no angle, an appearance which certainly implies that they are seen as curves

A person of Mr. Parsey's talent and reflection might make a valuable addition to the literature of art, by a work on the subject to which we have referred; but this is not the object of the work before us, which professes to teach the geometrical art commonly termed linear perspective, and we must own that we regard it as a failure.

The system recommended by the author was that generally employed, before Dr. Brook Taylor and Mr. Hamilton, towards the close of the last century, placed linear perspective on its legitimate foundation, and gave it all the precision and elegance which distinguish it. We think, therefore, that Mr. Parsey, in endeavouring to simplify the apparent difficulties of the practical application of the principles established by these masters, has made an innovation without any improvement, and has really sacrificed both accuracy and intelligibility.

He seems to overlook the fact, that an outline on paper ought to be the section by a plane of the cones of rays proceeding from the contours of an object by the eye; and that by taking the chords of the angles subtended by the original lines, the outline he deduces is essentially false, because he thereby assumes several unconnected planes of projection. For example, in his fifth plate the object is a cube, and he assumes the plane as not parallel to a face of the solid; in this case the upright lines of the cube would have a vanishing point, that is, the lines representing them would not be parallel as he has drawn them; in fact the whole of his construction is totally erroneous in this and several other instances, though the style of the diagrams prevents the defect from being immediately apparent.

We would earnestly advise any one desirous of acquiring the principles of perspective, to have recourse at once to writers who have treated the subject purely mathematically, and they may be assured that they will find no difficulties if they are conversant with plane and solid geometry; without that previous knowledge, they will never master the subject.

* We will take an opportunity of elucidating this, and some other points on the subject, in a future Number of this Journal.

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