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FMIEirS MATHEMATICAL DIAGRAMS ; PERIGAL’S LUXARIAX, Ac.

Class X.

Ward (No. 664, pp. 46G*, 467*) exhibits his botanicalcases, which are fully described in the Illustrated Cata-logue. A Prize Medal was awarded to him.

De la Rue (Class XVII., No. TO, p. 542) has exhibitedvarious applications of iridescent films for the purposes ofdecoration, their vivid colours being produced simply bythe agency of light upon a thin, transparent film of var-nish. The process adopted to render the film and its re-flected colours permanent, together with the method ofits application, are as follows:—

The objects to be ornamented, whether insects, shells,birds, bronzes, paper-hangings, card-cases, Ac., are im-mersed iu a vessel of water. Upon the surface of thelatter, when perfectly tranquil, is dropped a little oil orspirit varnish, which, spreading in all directions, becomesexceedingly attenuated, and reflects the most vivid coloursof the spectrum. The varnish being fixed, the object,which is slowly raised in such manner that the film shalladhere to its surface, is then placed in a convenient situa-tion, to permit the water draining off. When completelydry the film is found to be firmly attached, and perfectlyiridescent, having lost nothing of its original brilliancy ofcolouring. This is a beautiful illustration of the produc-tion of colour on a thin transparent surface, by the slightagency of light, such as is transiently seen in an ordinarysoap-bubble. The Jury awarded a Prize Medal to Mr.De la Rue.

Van Sciikndel (Belgium, No. 173, p. 1157) exhibitsan exceedingly good illustration of the laws of perspective,consisting of a series of objects, of exaggerated forms anddimensions, painted upon a horizontal plane, and two ver-tical planes. At a certain height above the picture thereis a circular hole, cut in a small wooden frame; on look-ing through which, the objects assume a natural appear-ance, iu strict accordance with the rules of perspective, andappear in the most perfect relief. A Prize Medal wasawarded to Mr. Van Schendel .

Ftsiikr (United States, No. 203. p. 1452) has exhibited“ mathematics simplified,” consisting of some beautifully-drawn diagrams, intended to facilitate the study of ma-thematics. His idea is that of teaching a physicalgeometry, either preliminary to, or, when no bettermay be bad, instead of the science. His method ofusing the diagrams is by teaching each step by a courseof reasoning, and illustrating the laws by well-drawnn r . *s.

Is iltody can question the great disadvantage underwhich students lie who have to apply geometry graphi-cally, if their previous figures have been drawn only byhand; or, what is worse, if badly drawn by ruler andcompasses. It is doubtful which of the two is the greaterevil—the giving a student ruler and compass as part ofhis course of geometry, or the making reasoning on badly-drawn figures the only preparation for a draughtsman orarchitect, Ac. This has been often said, but seldom ac-companied with any proof of the very satisfactory usewhich may be made of well-drawn diagrams. Mr. Fisherhas the merit of offering this proof to the Exhibition,with some ingenious ideas as to the manner in which thedetails may be managed. The attention he has paid toone poiut, viz., the exhibition of areas of given simpleratios under the same and different forms is particularlybeneficial. Some of his diagrams are, in this respect, ex-cellent studies for an eye which is to be trained to correctestimation. His method is particularly applicable foradults possessed of a power of thought, which requires tohe enlisted and exercised to make their study agreeableor even profitable.

Mr. Fisher’s merit may be described as consisting in—1st, the application of the idea of teaching by physicalperception, to a wider range of subject than merely makingvery exact drawings of the propositions to be demon-strated in Euclid ; 2nd, iu the ingenuity of his details ;3rd, the beauty of the drawings. The Jury consideredMr. Fisher deserving Honourable Mention.

Perigal (No. 593, p. 472*) has exhibited a demonstra-tion, by the transposition of parts, of the theorem of theright-angled triangle. The square or base of a right-angled triangle being intersected by two straight lines,passing through its centre, parallel and perpendicular to

the hypothenuse, is thereby divided into four parts equaland similar to each other : which, being symmetricallyarranged around a square equal to that on the perpendi-cular, form therewith a square equal to that on the hypo-thenuse of the right-angled triangle conversely. Thesides of the hypothenuse square being bisected, and thepoints of the bisecting lines being drawn ■ till they meet)parallel to the base and to the perpendicular, the hvpo-thenuse square is thereby divided into four equal quadri-laterals, which are together equivalent to the square orthe base, encompassing a square equal to that on the per-pendicular. Thus proving that “ in a right-angled tri-angle the square of each side is equivalent to the sum, orto the difference , of the squares of the other two sides.”—Euclid , i. 47.

Mr. Perigal has also exhibited a quadratic trisection ofthe square. The square is divided into nine parts ofthree different shapes and sizes ; so proportioned that, bycombining together one of each of them, they will formthree equal squares, each one-third the area of the squareformed by the whole nine sections. The construction isscarcely susceptible of brief explanation without adiagram.

Mr. Perigal likewise exhibits diagrams of the retrogres-sive parabola, as derived from the circle, // = R cos <p;x — R cos 2 <p = R (2 cos® p — 1) . *. 2 = R (R -j- x) t

The origin of co-ordinates at centre of circle, radius R.The kinematic curve, of which the retrogressive parabolais a limit, was discovered by Mr. Perigal, in 1835, andproduced from continuous motion by him in 1840.

Mr. Perigal has also exhibited a lunarian, of novelconstruction, which he calls a “ selenescopeintended toelucidate the kinematic effects of the three hypotheseswhich have been advanced to account for the inhabitantsof the earth never having seen more than one-half thesurface of the moon, the same hemisphere of our satellitebeing always presented towards the earth. For thispurpose, a terrestrial globe (about 3 inches diameter) isfixed on a brass stem supported by a brass pedestal. Atthe bottom of the stem is a fixed wheel of forty-eightteeth; and, between that and the pedestal, a T-shapedbrass arm is centred, having at one end a receptacle fora mariner’s compass. The other extremity of the arm isgrooved to receive four arbors with each a wheel of forty-eight teeth like the fixed wheel first mentioned, intowhich one of them is made to gear. Another wheel inthe middle of the cross part of the T gears into thesecond, and a fourth at the extremity into the third;while the fifth occupies a position at the opposite end ofthe cross groove, but does not gear with the other wheels.All these wheels have the same number of teeth. To thethird, fourth, and fifth wheels are attached spindles, eaclicarrying an ivory ball (about one inch diameter), repre-senting the moon, at such elevation that their centresand that of the globe, representing the earth, may be allin the horizontal plane.

Carried by the arm round the fixed wheel, into whichit gears, the second wheel is constrained to turn round itsaxis or arbor in the same direction, driving the thirdwheel in the contrary direction, by which the fourthwheel is driven in the same direction as that of therevolving arm; all with the same angular velocity:while the fifth wheel, not being in gear with the others,revolves with the arm without any additional rotation onits own arbor or axis. A coloured spot upon each of theivory moons tends to render their relative motions moreperceptible and distinct.

By this means one of the moons is caused to rotate onits axis in the same time, and in the same direction inwhich it revolves; another is caused to rotate on its axisin the same time in the contrary direction to which itrevolves; while the third moon revolves about the earth,but does not rotate round its own axis.

Mr. Perigal also exhibited a gyroscope; an instrumentdesigned to illustrate the effects of revolution and rota-tion. On a brass pedestal, an arm, supported at one endby an axis round which it freely revolves, carries at itsother extremity a globe (about one inch diameter) repre-senting the earth or the moon ; which is flange-jointed inthe middle, in order that the globe may be placed over