Volume 2

A mathematical and philosophical dictionary: containing an explanation of the terms, and an account of the several subjects, comprized under the heads mathematics, astronomy, and philosophy both natural and experimental: with an historical account of the rise, progress, and present state of these sciences: also memoirs of the lives and writings of the most eminent authors. Both ancient and modern, who by their discoveries or improvements have contributed to the advancement of them ... With many cuts and copper-plates / By Charles Hutton.

  • Hutton, Charles, 1737-1823.
Date:
1795-1796
Catalogue details

Licence: Public Domain Mark

Credit: A mathematical and philosophical dictionary: containing an explanation of the terms, and an account of the several subjects, comprized under the heads mathematics, astronomy, and philosophy both natural and experimental: with an historical account of the rise, progress, and present state of these sciences: also memoirs of the lives and writings of the most eminent authors. Both ancient and modern, who by their discoveries or improvements have contributed to the advancement of them ... With many cuts and copper-plates / By Charles Hutton. Source: Wellcome Collection.

    22/818 (page 14)
    traaed towards the perpendicular EA; and fo likewifc vrill the ray incident at B ; in confequence of which, they will diverge from one another within the denfe me- dium. T. he-intermediate rayswill alfo diverge more or lefs, as they recede from the axis TCj which, be- ' ing in the perpendicular, will go draight on. ^If the rays be parallel within*the denfe medium, they «will diverge when they pafs from thence into a rarer rnedivim, through a concave furface. For the ray DA will be refracted from the perpendicular AE, as will alfo the ray that is incident at B, together with all the intermediate rays, in proportion to their diilance from .theaxis or central ray TC. Therefore, if a denfe medium, as the glafs AB, be terminated by two concave furfaces, parallel rays pafT- ing through it will be made to diverge by both the fides of it. Thus the firil furface AB will make them diverge as if they had come from the point T j and with the effecl of the fecond furface added to this, they will diverge as from a nearer point, F. It was Kepler, v^'^ho by tb.efe inveftigations Gril gave a clear explanation of the effefls of lenfes, in making the rays of a pencil of light converge or diverge. Pic fhewed that a plano-convex lens makes rays, that were paiallel to its axis, meet at the didance of the diameter of the fphere of convexity j but that if both Tides of the lens be equally convex, the rays will have their fo- cus at the diilance of the radius of the circle corre- fpondingto that degree of convexity. But he did not inveftigate any rule for the foci ofienfes unequally con- vex. He only fays, in general, that they will fall fome- w'herc in the medium, between the foci belonging to the two different degrees of convexity. It is to Cavalerius that we owe this invelb’gation ; he laid down this rule, As the fum of both the diameters is to one of them, fo is the other to the diflance of the focus. And it is to be noted that all thefe rules, concerning convex lenfes, are applicable 10 thofe that are concave, with this dif- ference, that the focus is on the contrary fide of the glafs. See Montucia, vol. 2, pa, 17b : or Prielllev’s Plift. of Vifion, pa. 65, 410. Upon this principle it v/as not difficult to find the fpei of pencils ofrays iffuing from any point in the axis of the lens.; fmee thofe that are parallel will meet in the focus; and if they iffue from the focus, they will be parallel on the other fide. If they iffue from a point between the focus and the glafs, they will continue to diverge after paffing the lens, but lefs than before ; while thofe that come from beyond the focus, will converge after palling the glafs, and will meet in a place beyond the oppofite focus. This philofopher particularly ob- ferved, that rays which ifTue from twice the diilance of the focus, will meet at the fame diilance on the other fide. The moll important of thefe obfervations have been already illuftrated by proper figures, and from them the reft may be eafily conceived. Later optical writers have affigncd the diliances at which rays will meet, that iffue from any other place in the axis of a lens; bat Keple r was too much intent upon his allro- nomical and other purfuits, to give much attention to geometry. But, from the whole, Montuc-la gives the following rule concerning this fubjedl; As the excels of the diilance of the obje6l from the glafs, above the diilance of the locus, is to the diilance of the focus ; fo is this diflance, to the place of convergency beyond tne glafs. And the fame rule will find the point of divergency, when the rays iffue from any place between the lens and the focus ; for then the excefs of the dif- tance of the object from the glafs, above that of the/ focus, is negative, which is the lame diilance taken the contrary way. Montucia, vol. 2, pa. 177. And from the principle above-mentioned, it will not be dilliicult to iinderlland the application of lenfes, in the rationale of telefcopes and microfcopes. On thefe prin- ciples too is founded the ffrudlure of refradling burning glaffes, by whicn tiie fun^s light and heat are exceed- ingiy augmented in the focus of the lens, whether con- vex or plano-convex; fince the rays, falling parallel to the axis of the lens, are reduced into a much narrower compafs ; fo that it is no wonder they burn fome bo- dies, melt others, and produce other extraordinary phe- nomena. In the Philof. T 'ranf. vol. xvii. 960, or the Abr. i, 191, Dr. Halley gives an ingenious invelligation of the foci of rays refraded through any lenfes, nearly as follows ! Let BEL be a double convex lens, C the centre of the fegment EB, and K the centre of the fegrnent EL; BL the thicknefs or diameter of the lens, and D a point in the axis ; it is required to find the point F, or focus, w'here the rays proceeding from D lhal! be col- lected, after being- rcfraCled through the lens at A and a, points very near to the axis BL. Put the diilance I)A or the radius CA or CB=rr, and the ra- dius Ka or KL=r:R ; alfo the thicknefs of the lens and m to n the ratio of the fine of the angle of incidtnice DAG to the fine of the refraded angle HAG or CAM ; or m to n will be the ratio of thofe angles themfelves nearly, fince very fmall angles are fo each other in tlie fame ratio as their fines. Hencef w is as the angle DAG or DAC, n is as the angle H AG or MAC, and becaufe in this cafe the fides are as thclArpno. , fite