Elements of philosophy the first section, concerning body / written in Latine by Thomas Hobbes of Malmesbury ; and now translated into English ; to which are added Six lessons to the professors of mathematicks of the Institution of Sr. Henry Savile, in the University of Oxford.

CHAP. XXIV. Of Refraction and Reflection.

• 1 Definitions.
• 2 In perpendicular Motion there is no Refraction.
• 3 Things thrown out of a thinner into a thicker Medium, are so refracted, that the Angle Refracted is greater then the Angle of Inclination.
• 4 En∣deavour which from one point tendeth every way, will be so Refracted, at that the sine of the Angle Refracted, will be to the sine of the Angle of Inclination, as the Density of the first Medium is to the Density of the second Medium, reciprocally taken.
• 5 The sine of the Refracted Angle in one Inclination is to the sine of the Refracted Angle in another Inclination, as the sine of the Angle of that Inclination is to the sine of the Angle of this Inclination.
• 6 If two lines of Incidence, having equal Inclination, be the one in a thinner, the other in a thicker Medium, the sine of the angle of Inclina∣tion will be a Mean proportional between the two sines of the Refracted angles.
• 7 If the angle of Inclination be semirect, and the line of Inclination be in the thicker Medium, and the proportion of their Densities be the same with that of the Diagonal to the side of a Square, and the separating Superficies be plain, the Refracted line will be in the separating Superficies.
• 8 If a Body be carried in a straight line upon another Body, and do not penetrate the same, but be reflected from it, the angle of Reflexion will be equal to the Angle of Incidence.
• 9 The same happens in the generation of Motion in the line of Incidence.

1

Definitions.

• 1 REFRACTION, is the breaking of that straight Line, in which a Body is moved, or its Action would proceed in one and the same Medium, into two straight lines, by reason of the different natures of the two Mediums.
• 2 The former of these is called the Line of Incidence; the later the Refracted Line.
• 3 The Point of Refraction, is the common point of the Line of In∣cidence and of the Refracted Line.
• ...

• 4 The Refracting Superficies, which also is the Separating Superficies of the two Mediums, is that in which is the point of Refra∣ction.
• 5 The Angle Refracted, is that which the Refracted Line makes in the point of Refraction, with that Line which from the same point is drawn perpendicular to the separating Superfi∣cies in a different Medium.
• 6 The Angle of Refraction, is that which the Refracted line makes with the Line of Incidence produced.
• 7 The Angle of Inclination, is that which the Line of Incidence makes with that Line which from the point of Refraction is drawn perpendicular to the separating Superficies.
• ...

  8 The Angle of Incidence, is the Complement to a right Angle of the Angle of Inclination.

  And so, (in the first Figure) the Refraction is made in A B F. The Refracted Line is B F. The Line of Incidence is A B. The Point of Incidence, and of Refraction is B. The Refracting or Separating Su∣perficies is D B E. The Line of Incidence produced directly is A B C The Perpendicular to the separating Superficies is B H. The Angle of Refraction is C B F. The Angle Refracted is H B F. The Angle of Inclination is A B G or H B C. The Angle of Inci∣dence is A B D.

• 9 Moreover the Thinner Medium, is understood to be that in which there is less resistance to Motion or to the generation of Motion; & the Thicker, that wherin there is greater resistance.
• 10 And that Medium in which there is equal resistance every where, is a Homogeneous Medium. All other Mediums are Hete∣rogeneous.

2 If a Body pass, or there be generation of Motion, from one Medium to another of different Density, in a line perpendicular to the Separating Superficies; there will be no Refraction.

For seeing on every side of the perdendicular all things in the Mediums are supposed to be like and equal; if the Motion it self be supposed to be perpendicular, the Inclinations also will be e∣qual, or rather none at all; and therefore there can be no cause, from which Refraction may be inferred to be on one side of the

perpendicular, which wil not cōclude the same Refraction to be on the other side. Which being so, Refraction on one side will destroy Refraction on the other side; and consequently, either the Refra∣cted line will be every where, (which is absurd), or there will be no Refracted line at all; which was to be demonstrated.

Corol. It is manifest from hence, that the cause of Refraction con∣sisteth onely in the obliquity of the line of Incidence, whether the Incident Body penetrate both the Mediums, or without penetra∣ting, propagate motion by Pressure onely.

3 If a Body, without any change of situation of its internal parts, as a stone, be moved obliquely out of the thinner Medium, and proceed penetrating the thicker Medium; and the thicker Me∣dium be such, as that its internal parts being moved, restore them∣selves to their former situation; the angle Refracted will be greater then the angle of Inclination.

For let D B E (in the same first figure) be the separating Super∣ficies of two Mediums; and let a Body, as a stone thrown, be un∣derstood to be moved as is supposed in the straight line A B C; and let A B be in the thinner Medium, as in the Aire; and B C in the thicker, as in the Water. I say the stone, w^ch being thrown, is moved in the line A B, will not proceed in the line B C, but in some other line, namely that, with which the perpendicular B H makes the Refracted angle H B F greater then the angle of Inclination H B C.

For seeing the stone coming from A, and falling upon B, makes that which is at B proceed towards H, and that the like is done in all the straight lines which are parallel to B H; and seeing the parts moved restore themselves by contrary motion in the same line; there will be contrary motion generated in H B, and in all the straight lines which are parallel to it. Wherefore the motion of the stone will be made by the concourse of the motions in A G, that is, in D B, and in G B, that is, in B H, and lastly, in H B, that is, by the concourse of three motions. But by the concourse of the motions in A G and B H, the stone will be carried to C; and therefore by ad∣ding the motion in H B, it will be carried higher in some other line, as in B F, and make the angle H B F greater then the angle H B C.

And from hence may be derived the cause, why Bodies which

are thrown in a very oblique line, if either they be any thing flat, or be thrown with great force, will when they fall upon the water, be cast up again from the water into the aire.

For let A B (in the 2d figure) be the superficies of the water; in∣to which from the point C, let a stone be thrown in the straight line C A, making with the line B A produced a very little angle C A D; and producing B A indefinitely to D, let C D be drawn perpendi∣cular to it, and A E parallel to C D. The stone therefore will be moved in C A by the concourse of two motions in C D and D A, whose velocities are as the lines themselves C D and D A. And from the motion in C D and all its parallels downwards, as soon as the stone falls upon A, there will be Reaction upwards, be∣cause the water restores it self to its former situation. If now the stone be thrown with sufficient obliquity, that is, if the straight line C D be short enough, that is, if the endeavour of the stone downwards be less then the Reaction of the water upwards, that is, less then the endeavour it hath from its own gravity, (for that may be), the stone will (by reason of the excess of the endeavour which the water hath to restore it self, above that which the stone hath downwards) be raised again above the Superficies A B, and be carried higher, being reflected in a line which goes higher, as the line A G.

4 If from a point, whatsoever the Medium be, Endeavour be propagated every way into all the parts of that Medium; and to the same Endeavour there be obliquely opposed another Medium of a different nature, that is, either thinner or thicker; that Endea∣vour will be so refracted, that the sine of the angle Refracted, to the sine of the angle of Inclination, will be as the density of the first Medium to the density of the second Medium, reciprocally ta∣ken.

First, let a Body be in the thinner Medium in A (Figure 3d.); and let it be understood to have endeavour every way, and conse∣quently that its endeavour proceed in the lines A B and A b; to which let B b the superficies of the thicker Medium be obliquely opposed in B and b, so that A B and A b be equal; and let the straight line B b be produced both wayes. From the points B and b let the perpendiculars B C and b c be drawn; and upon the centers

B and b, and at the equal distances B A and b A, let the Circles A C and A c be described, cutting B C and b c in C and c, and the same C B and c b produced in D and d, as also A B and A b pro∣duced in E and e. Then from the point A to the straight lines B C and b c let the perpendiculars A F and A f be drawn. A F therefore will be the sine of the angle of Inclination of the straight line A B, and A f the sine of the angle of Inclination of the straight line A h, which two Inclinations are by construction made equal. I say, as the density of the Medium in which are B C and b c, is to the density of the Medium in which are B D and b d, so is the sine of the angle Refracted, to the sine of the angle of Inclination.

Let the straight line F G be drawn parallel to the straight line A B, meeting with the straight line b B produced in G.

Seeing therefore A F and B G are also parallels, they will be e∣qual; and consequently, the endeavour in A F is propagated in the same time, in which the endeavour in B G would be propagated if the Medium were of the same density. But because B G is in a thicker Medium, that is, in a Medium which resists the endeavour more then the Medium in which A F is, the endeavour will be propagated less in B G then in A F, according to the propor∣tion which the density of the Medium in which A F is, hath to the density of the Medium in which B G is. Let therefore the density of the Medium in which B G is, be to the density of the Medium in which A F is, as B G is to B H; and let the measure of the time be the Radius of the Circle. Let H I be drawn parallel to B D, meeting with the circumference in I; and from the point I let I K be drawn perpendicular to B D; which being done, B H and I K will be equal; and I K will be to A F, as the density of the Medium in which is A F, is to the density of the Medium in which is I K. Seeing therefore in the time A B (which is the Radius of the Circle) the endeavour is propagated in A F in the thinner Me∣dium, it will be propagated in the same time, that is, in the time B I in the thicker Medium from K to I. Therefore B I is the Refra∣cted line of the line of Incidence A B; and I K is the sine of the angle Refracted; and A F, the sine of the angle of Inclination. Wherefore seeing I K is to A F, as the density of the Medium in which is A F to the density of the Medium in which is I K; it will be as the density of the Medium in which is A F, (or

B C) to the density of the Medium in which is I K (or B D), so the sine of the angle Refracted to the sine of the angle of Inclination. And by the same reason it may be shewn, that as the density of the thinner Medium is to the density of the thicker Medium, so will K I the sine of the angle Refracted be to A F the sine of the Angle of Inclination.

Secondly, let the Body which endeavoureth every way, be in the thicker Medium at I. If therefore both the Mediums were of the same density, the endeavour of the Body in I B would tend di∣rectly to L; and the sine of the angle of Inclination L M would be equal to I K or B H. But because the density of the Medium in which is IK, to the density of the Medium in which is L M, is as BH to B G, that is, to A F, the endeavour will be propagated further in the Medium in which L M is, then in the Medium in which I K is, in the proportion of density to density, that is, of M L to A F. Wherefore B A being drawn, the angle Refracted will be C B A, and its sine A F. But L M is the sine of the angle of Inclination; and therefore again, as the density of one Medium is to the densi∣ty of the different Medium, so reciprocally is the sine of the angle Refracted to the sine of the angle of Inclination, which was to be demonstrated.

In this Demonstration, I have made the separating Superficies B b plain by construction. But though it were concave or convex, the Theoreme would nevertheless be true. For the Refraction be∣ing made in the point B of the plain separating Superficies, if a crooked line, as P Q be drawn, touching the separating line in the point B; neither the Refracted line B I, nor the perpendicular B D will be altered; and the Refracted angle K B I, as also its sine K I will be still the same they were.

5 The sine of the angle Refracted in one Inclination, is to the sine of the angle Refracted in another Inclination, as the sine of the angle of that Inclination to the sine of the angle of this Incli∣nation.

For seeing the sine of the Refracted angle is to the sine of the angle of Inclination, (whatsoever that Inclination be) as the density of one Medium, to the density of the other Medium; the proporti∣on of the sine of the Refracted angle, to the sine of the angle of In∣clination, will be compounded of the proportions of density to den∣sity,

and of the sine of the angle of one Inclination to the sine of the angle of the other Inclination. But the proportions of the densities in the same Homogeneous Body, are supposed to be the same. Wherefore Refracted angles in different Inclinations, are as the sines of the angles of those Inclinations; which was to be demon∣strated.

6 If two lines of Incidence having equal inclination, be the one in a thinner, the other in a thicker Medium; the sine of the angle of their Inclination, will be a mean proportional between the two sines of their angles Refracted.

For let the straight line AB (in the same 3d figure) have its Incli∣nation in the thinner Medium, and be refracted in the thicker Me∣dium in B I; and let E B have as much Inclination in the thicker Medium, and be refracted in the thinner Medium in B S; and let R S the sine of the angle Refracted be drawn. I say the straight lines R S, A F and I K are in continual proportion. For it is, as the density of the thicker Medium to the density of the thinner Me∣dium, so R S to A F. But it is also, as the density of the same thicker Medium, to that of the same thinner Medium, so AF to IK. Where∣fore R S. A F : : A F. I K are propoortionals; that is, R S, A F and I K are in continual proportion, and A F is the Mean proportio∣nal; which was to be proved.

7 If the angle of Inclination be semirect, and the line of Incli∣nation be in the thicker Medium, and the proportion of the Densi∣ties be as that of a Diagonal to the side of its Square, and the separating Superficies be plain, the Refracted line will be in that separating Superficies.

For in the Circle A C (in the 4th figure) let the angle of Incli∣nation A B C be an angle of 45 degrees. Let C B be produced to the Circumference in D; & let C E (the sine of the angle •• B C) be drawn▪ to which, let B F be taken equal in the separating line B G. B C E F will therefore be a Parallelogram, & F E & B C, that is, F E and B G equal. Let AG be drawn, namely, the Diagonal of the Square whose side is B G; and it will be, as A G to E F, so B G to B F; & so (by supposition) the density of the Medium in which C is, to the density of the Medium in which D is; and so also the sine of the angle Refracted to the sine of the angle of Inclination.

Drawing therefore F D, & from D the line D H perpendicular to A B produced, DH will be the sine of the angle of Inclination. And seeing the sine of the angle Refracted is to the sine of the angle of Inclination, as the density of the Medium in which is C, is to the density of the Medium in which is D, that is, (by supposition) as A G is to F E, that is, as D H is to B G; and seeing D H is the sine of the angle of Inclination, B G will therefore be the sine of the angle Refracted. Wherefore B G will be the Refracted line, and lye in the plain separating Superficies; which was to be demonstrated.

Coroll. It is therefore manifest, that when the Inclination is greater then 45 degrees, as also when it is less, provided the densi∣ty be greater, it may happen that the Refraction will not enter the thinner Medium at all.

8 If a Body fall in a straight line upon another Body, and do not penetrate it, but be reflected from it, the angle of Reflexion will be equal to the angle of Incidence.

Let there be a Body at A (in the 5th figure), which falling with straight motion in the line A C upon another Body at C, passeth no further, but is reflected; and let the angle of Incidence be any angle, as A C D. Let the straight line C E be drawn, making with D C produced the angle E C F equall to the angle A C D; and let A D be drawn perpendicular to the straight line D F. Also in the same straight line D F let C G be taken equall to C D; and let the perpendicular G E be raised, cutting C E in E. This be∣ing done, the triangles A C D and E C G will be equall and like. Let C H be drawn equal and parallel to the straight line A D; and let H C be produced indefinitely to I. Lastly let E A be drawn, which will passe through H, and be parallel and equall to G D. I say the motion from A to C in the straight line of Inci∣dence AC, will be reflected in the straight line C E.

For the motion from A to C is made by two coefficient or con∣current motions, the one in A H parallel to D G, the other in A D perpendicular to the same D G; of which two motions, that in A H workes nothing upon the Body A after it has been moved as farre as C, because (by supposition) it doth not passe the straight line D G; whereas the endeavour in A D, that is in H C, work∣eth further towards I. But seeing it doth onely presse and not pe∣netrate,

there will be reaction in H, which causeth motion from C towards H; and in the mean time the motion in H E re∣maines the same it was in A H; and therefore the Body will now be moved by the concourse of two motions in C H and H E, which are equall to the two motions it had formerly in A H and H C. Wherefore it will be carried on in C E. The angle therefore of Re∣flection will be E C G, equall (by construction) to the angle A C D; which was to be demonstrated.

Now when the Body is considered but as a point, it is all one, whether the Superficies or line in which the Reflection is made, be straight or crooked; for the point of Incidence and Reflexion C, is as well in the crooked line which toucheth D G in C, as in D G it selfe.

9 But if we suppose that not a Body be moved, but some Endea∣vour onely be propagated from A to C, the Demonstration will neverthelesse be the same. For all Endeavour is motion; and when it hath reached the Solid Body in C, it presseth it, and endeavoureth further in C I. Wherefore the reaction will pro∣ceed in C H; and the endeavour in C H concurring with the en∣deavour in H E, will generate the endeavour in C E, in the same manner as in the repercussion of Bodies moved.

If therefore Endeavour be propagated from any point to the concave Superficies of a Spherical Body, the Reflected line with the circumference of a great circle in the same Sphere, will make an angle equall to the angle of Incidence.

For if Endeavour be propagated from A (in the 6 fig.) to the circumference in B, and the center of the Sphere be C, and the line C B be drawne, as also the Tangent D B E; and lastly if the angle F B D be made equall to the angle A B E, the Reflexion will be made in the line B F, as hath been newly shewn. Where∣fore the angles which the straight lines A B and F B make with the circumference, will also be equall. But it is here to be noted that if C B be produced howsoever to G, the endeavour in the line G B C will proceed onely from the perpendicular reaction in G B; and that therefore there will be no other endeavour in the point B towards the parts which are within the Sphere, besides that which tends towards the center.

And here I put an end to the third part of this Discourse; in which I have considered Motion and Magnitude by themselves in the abstract. The fourth and last part, concerning the Phaenomena of Nature, that is to say, concerning the Motions and Magnitudes of the Bodies which are parts of the World, reall and existent, is that which followes.