these questions it is endeavoured to find commenfurable numbers to answer indetermediate problems; which bring out an infinite number of incommenfurable quantities. See INDETERMINATE PROBLEMS. DIOPHANTUS, a mathematician of Alexandria, who is believed to have lived about the 3d century. We have his works, which were publifhed with notes at Paris, in 1621, by Bachet de Meziriac; and another edition in 1670, with observations on every queftion, by M. Fermat. DIOPTER, the index of an aftrolabe. DIOPTRA, in inftrument invented by Hippar chus, for levelling water courfes; taking the height of towers, and determining the distances of planets, &c. *DIOPTRICAL.} a medium for the fight; adj. [da] Affording DIOPTRICK. affifting the fight in the view of diftant objects.-Being excellently well furnished with dioptrical glaffes, he had been able to fee the fun fpotted. Boyle.-View the afperites of the moon through a dioptrick glafs, and venture at the proportion of her hills by their fhadows. More's "Antitode against Atheism. DIOPTRICS. INTRODUCTION. DEFINITIONS and HISTORY of DIOPTRICS. (1.) THIS branch of science, which is alfo called ANACLASTICS, is thus defined by Dr Johnfon: (2.) * DIOPTRICKS. n.. A part of opticks, treating of the different refractions of the light paffing through different mediums; as the air, water, glaffes, &c. Harris. (3.) DIOPTRICS, from A thro' and orla, to fee, may be more clearly defined, the doctrine of refracted vifion, or that part of optics, which explains the effects of light, as refracted by paffing through different mediums; fuch as air, water, glafs, &c. and especially lenfes. (4) This branch of optics is one of the moft ufeful and pleasant of the sciences, because by it we are enabled to view remote objects, as if they were near at hand, and such minute objects as without its affiftance would entirely clude our fight. The advantages which mankind have derived from this science, by the fimple invention of SPECTACLES, warrant us to fay, that it has in a manner given fight to the blind. (5.) The ancients have treated of direct and reflected vifion, but what they left upon refracted vifion is very imperfect. J. BAPTIST PORTA wrote a treatise on refraction in 9 books, but without any great improvement. (6.) KELPER was the firft who fucceeded in any great degree upon this fubject, having demonftrated very accurately the properties of fpherical lenfes, in a treatife first published in 1611. GALILEO gave fomewhat of this doctrine in his letters; as alfo an Examination of the Preface of JOHANNES PENA upon Euclid's Optics, concerning the u'e of optics in Aftronomy. (7.) DES CARTES alfo wrote a treatife on dioptrics, commonly annexed to his Principles of Philofophy; in which the true manner of vilion is more diftinely explained than by any former writer, and in which is contained the law of refraction discovered by SNELL, though the name of the inventor is fuppreffed. This work contains alfo the properties of elliptical and hyperbolic lenfes, with the practice of grinding glafies. (8.) Dr Barrow has treated on dioptrics in an elegant, though rather brief manner, in his Opti cal Lectures. There are alfo HUYGENS's Diop trics, an excellent work of its kind; MOLY. NEUX's Dioptrics, a work rather heavy and dull; HARTSOEKER'S Effai de Dioptrique : CHERUBIN'S Dioptrique Oculaire, et La Vifion Parfaite; DAVID GREGORY's Elements of Dioptrics; TRABER'S Nervus Opticus; ZAHN's Oculus Artificialis Teledioptricus, Dr SMITH'S Optics, a complete work of its kind; WOLFIUS's Dioptrics, &c. but above all, The Treatife on Optics, and the Optical Lectures of Sir I. NEWTON, in whofe experiments are contained far more difcoveries, than in all the former works on the fubject. (9.) This fcience was nearly perfected, by Dollond's discovery of the Achromatic Glaffes; which has been rendered ftill more complete by the experiments of Dr ROBERT BLAIR, related in the 3d volume of the Edin. Phil. Trans. (10.) As this fubject will be treated fully under the general article OFTICS, we shall here, as in the article CATOPTRICS, be very brief. SECT. I. DEFINITIONS of TERMS. (11.) DEF. I. When a ray of light paffing out of one medium into another of a different density, is turned from that ftraight line, in which it would otherwife proceed into one of a different direction it is faid to be REFRACTED. Thus if AB, a ray of light proceeding from the radiant point A, (Plate CVII. fig. 1.) fall obliquely upon FG, the furface of a transparent medium, either denfer or rarer than the medium along which the ray paffed from the point A; the ray will not proceed in the ftraight line ABC, but will have its direction bent into the line BD. This deviation is called the RE FRACTIONS OF THE RAY. (12.) II. AB is called the LINE OF INCIDENCE, and the ray which proceeds in the direction AB is called the INCIDENT RAY; BD is called the LINE OF REFRACTION, and the ray which proceeds in that direction is called the REFRACTED RAY. (13.) III. Let HBK be drawn perpendicular to the common furfaces of the contiguous mediums; the angle contained by AB, the line of incidence, and BH, the perpendicular, is called the ANGLE OF INCIDENCE; and the angle contained by BD, the line of refraction, and BK, the perpendicular, is called the ANGLE OF REFRACTION. (14.) IV. and F will appear; and when it rifes to M, allthe three pieces will be vifible. (14.) IV. Any spherical transparent glats, that converges or diverges the rays of light as they pafs through it, is called a LENS. (15.) V. There are five forts of lenfes; viz. 1. A plane or fingle convex lens, which is plane on one fide and convex on the other; as AZ, Ag. 2. 2. A double convex lens, as B. 3. A plano concave lens, that is, plane on one fide and concave on the other, as C. 4. A double concave, as D. And, 5. A menifcus, which is convex on one fide and concave on the other, as E. (16.) VI. The point C, round which the fpherical furface of a lens, as AZ, is defcribed, is called its CENTRE; the line XY, drawn from that centre perpendicular to its two furfaces, is the AXIS; and the point V, to which the axis is drawn, is the VERTEX of that lens. (17.) VII. When the rays of light that pafs through a fingle or double convex lens are brought into their smalleft compass, that point is the Focus of the lens. (18) VIII. In optical inftruments, that lens which is next the object is called the OBJECT GLASS; and that next the eye, the EYE GLASS. SECT. II. GENERAL PRINCIPLES of DIOPTRICS. (19.) THE general laws, by which the phenomena of refraction through lenfes may be explained, are as follow: (20) I. A ray of light in paffing obliquely out of a rarer medium into a denfer, e. g. out of air into water or glafs, is refracted towards the perpendicular. Thus if the medium above FG, fig. 1. be air, and the medium below it water, the angle ABH will be greater than KBD. (21.) II. The fines of the angles of incidence and refraction have a conftant ratio to one another. When a ray pafies out of air into water, the fine of the angle of incidence is to the fine of the angle of refraction, as 4 to 3; and when the ray paffes out of air into glafs, they are to one another as 3 to 2. (22.) III. When a ray paffes out of a denfer into a rarer medium, it is refracted from the perpendicular. (23. IV. A ray falling on a curve furface, whether concave or convex, is refracted after the fame manner as if it fell on a plane, which is a tangent to the curve at the point of incidence. (24.) For the various phenomena of refraction by lentes, fee OPTICS. SECT. III. ENTERTAINING EXPERIMENTS by OPTICAL ILLUSIONS. (25.) I. On the bottom of the veffel ABCD, Fig. 3. place three pieces of money; as a thilling at E, a half-crown at F, and crown at G. Then place a perfon at H, where he can fee no further into the veffel than I and tell him, that by pouring water into the vellel you will make him fee three different pieces of money; bidding him obferve carefully whether any money goes in with the water. The water muft be poured in very gently, or the pieces fo fixed, that they may not move out of their places by its agitation. When the water comes up to K, the piece at E will become visible; when it comes up to L, the pieces at E (26.) From what has been faid of the refraction of light, the caufe of this phenomena will be evident: for while the veffel is empty, the ray HL will naturally proceed in a straight line: but in proportion as it becomes immerfed in water; it will be neceffarily refracted into the feveral directions NE, OF, PG, and confequently the feveral pieces muft become vifible. (27.) II. OPTICAL AUGMENTATION. Take s large drinking glass of a conical figure, that is fmall at bottom and wide at top; in which put a fhilling, and fill the glafs about half full of water: then place a plate on the top of it, and turn it quickly over, that the water may not get out. A piece of the fize of a half-crown will then be feen on the plate, and fomewhat higher up, another piece of the fize of a fhilling. (28.) This phenomenon arifes from feeing the piece through the conical furface of the water at the fide of the glass, and through the flat surface at the top of the water; at the fame time: for the conical furface dilates the rays, and makes the piece appear larger; but by the flat furface the rays are only refracted, by which the piece is feen higher up in the glafs, but ftill of its natural fize. That this is the caufe will be further evident by filling the glafs with water; for as the fhilling cannot be then feen from the top, the large piece only will be visible. (29) III. OPTICAL SUBTRACTION. Against the wainscot of a room fix three fmall pieces of paper, as A, B, C, Fig. 4 at the height of your eye; and placing yourself directly before them, fhut your right eye and look at them with the left; when you will fee only two of those papers, fuppofe A and B; but altering the position of your eye, you will then fee the third and one of the firft, fuppofe A; and by altering your pofition a fecond time, you will fee B and C; but never ail the three together. (30.) The cause of this phenomenon is, that one of the three pencils of rays that come from these objects, falls conftantly on the optic nerve at D; whereas to produce diftin&t vifion, it is necessary that the rays of light fall on fome part of the retina, E, F, G, H. We fee by this experiment, one of the ufes of having two eyes; for he that has one only, can never fee three objects placed in this position, nor all the parts of one object of the fame extent, without altering the fituation of his eye. (31) IV. ALTERNATE ILLUSION. With a convex lens of about an inch focus, look attentively at a filver feal, on which a cipher is engraved. It will at first appear cut in, as to the naked eye: but if you continue to obferve it some time, without changing your fituation, it will feem to be in relief, and the lights and thades will appear the fame as they did before. If you regard it with the fame attention still longer, it will again appear to be engraved and fo on alternately. 32.) If you look off the feal for a few moments when you view it again, inftead of feeing it, as at firft, engraved, it will appear in relief. If, while you you are turned toward the light, you fuddenly incline the feal, while you continue to regard it, thofe parts that feemed to be engraved will immediately appear in relief; and if, when you are regarding thefe feeming prominent parts, you turn yourself fo that the light may fall on the right hand, you will fee the thadows on the fame fide from whence the light comes, which will appear not a little extraordinary. In like manner the fhadows will appear on the left, if the light fall on that fide. If, inftead of a feal, you look at a piece of money, thefe alterations will not be vifible, in whatever fituation you place yourself. (33.) It has been fufpected that this illufion, arifes from the fituation of the light: and in fact, "I have obferved (fays M. GUYOT, from whom this article is taken), that when I have viewed it with a candle on the right, it has appeared en graved; but by changing the light to the left fide it has immediately appeared in relief." It ftill however, remains to be explained, why we fee it alternately hollow and prominent, without changing either the fituation or the light. Perhaps it is in the fight itlelf that we must look for the caufe of this phenomenon ; and this feems the more probable, as all these appearances are not difcernible by all perfons. (34) Mr William Jones of Holborn has remarked, that this illufion is still more extrordinary and permanent, when one looks at a cavity in a feal or other object through the three eye-glaffes of a common four glafs 'refracting telescope: all cavities viewed through thefe glaffes appear conftant ly reliefs, in almost all fituations of the light. (35) V. The DIOPTRICAL PARADOX. A new and curious optical, or more properly a DIOPTRICAL deception, has been made by Mr W. Jones. Its effect is, that a print, or an ornamented draw. ing, with any object, fuch as an ace of diamonds, &c, in the centre F, fig. 5 will be feen as the ace of clubs, when it is placed in the machine ABCD, and viewed through a fingle glass only contained in the tube E. The conftruction of this machine is truly fimple. The glafs within the tube which brings about this furprifing change, is fomewhat on the principle of the common multiplying glafs, as reprefented at G, which by the number of its inclined furfaces and from the refractive power of the rays proceeding from the objects placed before it shows it in a multiplied ftate or quantity. Its only difference is, that the fides of this glafs are flat, and diverge upwards from the base to a point in the axis of the glafs like a cone: the num ber of the fides is fix; and each fide, from its angular position to the eye, has the property of refracting from the border of the print F fuch a portion of it (defignedly there placed), as will make a part in the compofition of the figure to be represented: for the hexagonal and conical figure of this glass prevents any fight of the ace of diamonds in the centi: being feen; confequently the ace of clubs being previously and mechanical ly drawn in the circle of refraction in fix different parts of the border, at, 1, 2, 3, 4, 5, 6, and artfully disguised in the o namental border by blending them with it, the glafs in the tube at E will change the appearance of the ace of diamonds F into the ace of clubs H. In the fame manner may VOL. VII. PART I. other prints undergo fimilar changes, according to the will of any ingenious draughtsman who may defign them. The figure of the glass is clearly fhown at 1. 1 SECT. IV. Of the CAMERA OBSCURA, or DARK CHAMBER. (36.) MAKE a circular hole in the fhutter of a window, from whence there is a profpect of the fields or ftreet, or any other object not too near and in this hole place a convex glafs, either double or fingle, whofe focus is at the diftance of 5 or 6 feet. The diftance should not be less than three feet; for if it be, the images will be too small and there will not be fufficient room for the spectators. On the other hand, the focus fhould ne ver be more than 15 or 20 feet, for then the ima ges will be obfcure, and the colouring faint. The beft diftance is from 6 to 12 feet. (37.) Take care that no light enter the room but by this glafs; at a diftance from it, equal to that of its focus, place a pafteboard, covered with the whiteft paper: which should have a black bor der to prevent any of the fide rays from difturbing the picture. Let it be two feet and a half long and 18 or 20 inches high: bend the lengthi of it inwards, to form a part of a circle, whofe diameter is equal to double the focal distance of the glass. Then fix it on a frame of the fame figure, and put it on a moveable foot, that it may be eafily fixed at that exact distance from the glafs where the objects paint themselves to the greatest perfection. (38) When it is thus placed, all the objects that are in the front of the window will be paint ed on the paper, in an inverted position, with the greateft regularity and in the most natural colours. If a moveable mirror be placed without the window, by turning it more or lefs, all the objects that are on each fide of the window will appear on the paper. If instead of placing the mirror without the window you place it in the room, and above the hole (which must then be made near the top of the fhutter), you may receive the reprefentation on a paper placed horizontally on a table; and draw, at your leifure, all the objects that are there painted. (39) Nothing can be more pleafing than this experiment, especially when the objects are ftrongly enlightened by the fun: and not only land prof pects, but a fea port, when the water is fomewhat agitated, or at the fetting of the fun, prefents a very delightful appearance. This repre fentation affords the most perfect model for painters, as well for the tone of colours, as that degrada tion of fhades, occafioned by the interpofition of the air, which has been so justly expressed by some modern painters. (49.) The paper must have a circular form; for otherwife, when the centre of it is in the focus of the glafs, the two fides will be beyond it, and confequently the image will be confufed. If the frame were contrived of a fpherical figure, and the glafs were in its centre, the reprefentation would be ftill more accurate. If the object without be at the diftance of twice the focal length of the glafs, the image in the room will be of the fame magnitude with the objec&, 94 (41.) There |