Obs. 8. I caused the edges of two Knives to be ground truly strait, and pricking their points into a Board so that their edges might look towards one another, and meeting near their points contain a rectilinear Angle, I fasten'd their Handles together with Pitch to make this Angle invariable. The distance of the edges of the Knives from one another at the distance of four Inches from the angular Point, where the edges of the Knives met, was the eighth part of an Inch; and therefore the Angle contain'd by the edges was about one Degree 54: The Knives thus fix'd together I placed in a beam of the Sun's Light, let into my darken'd Chamber through a Hole the 42d Part of an Inch wide, at the distance of 10 or 15 Feet from the Hole, and let the Light which passed between their edges fall very obliquely upon a smooth white Ruler at the distance of half an Inch, or an Inch from the Knives, and there saw the Fringes by the two edges of the Knives run along the edges of the Shadows of the Knives in Lines parallel to those edges without growing sensibly broader, till they met in Angles equal to the Angle contained by the edges of the Knives, and where they met and joined they ended without crossing one another. But if the Ruler was held at a much greater distance from the Knives, the Fringes where they were farther from the Place of their Meeting, were a little narrower, and became something broader and broader as they approach'd nearer and nearer to one another, and after they met they cross'd one another, and then became much broader than before.
Whence I gather that the distances at which the Fringes pass by the Knives are not increased nor alter'd by the approach of the Knives, but the Angles in which the Rays are there bent are much increased by that approach; and that the Knife which is nearest any Ray determines which way the Ray shall be bent, and the other Knife increases the bent.
Obs. 9. When the Rays fell very obliquely upon the Ruler at the distance of the third Part of an Inch from the Knives, the dark Line between the first and second Fringe of the Shadow of one Knife, and the dark Line between the first and second Fringe of the Shadow of the other knife met with one another, at the distance of the fifth Part of an Inch from the end of the Light which passed between the Knives at the concourse of their edges. And therefore the distance of the edges of the Knives at the meeting of these dark Lines was the 160th Part of an Inch. For as four Inches to the eighth Part of an Inch, so is any Length of the edges of the Knives measured from the point of their concourse to the distance of the edges of the Knives at the end of that Length, and so is the fifth Part of an Inch to the 160th Part. So then the dark Lines above-mention'd meet in the middle of the Light which passes between the Knives where they are distant the 160th Part of an Inch, and the one half of that Light passes by the edge of one Knife at a distance not greater than the 320th Part of an Inch, and falling upon the Paper makes the Fringes of the Shadow of that Knife, and the other half passes by the edge of the other Knife, at a distance not greater than the 320th Part of an Inch, and falling upon the Paper makes the Fringes of the Shadow of the other Knife. But if the Paper be held at a distance from the Knives greater than the third Part of an Inch, the dark Lines above-mention'd meet at a greater distance than the fifth Part of an Inch from the end of the Light which passed between the Knives at the concourse of their edges; and therefore the Light which falls upon the Paper where those dark Lines meet passes between the Knives where the edges are distant above the 160th part of an Inch.
For at another time, when the two Knives were distant eight Feet and five Inches from the little hole in the Window, made with a small Pin as above, the Light which fell upon the Paper where the aforesaid dark lines met, passed between the Knives, where the distance between their edges was as in the following Table, when the distance of the Paper from the Knives was also as follows.
Distances of the Paper
And hence I gather, that the Light which makes the Fringes upon the Paper is not the same Light at all distances of the Paper from the Knives, but when the Paper is held near the Knives, the Fringes are made by Light which passes by the edges of the Knives at a less distance, and is more bent than when the Paper is held at a greater distance from the Knives.
Fig. 3.
Obs. 10. When the Fringes of the Shadows of the Knives fell perpendicularly upon a Paper at a great distance from the Knives, they were in the form of Hyperbola's, and their Dimensions were as follows. Let CA, CB [in Fig. 3.] represent Lines drawn upon the Paper parallel to the edges of the Knives, and between which all the Light would fall, if it passed between the edges of the Knives without inflexion; DE a Right Line drawn through C making the Angles ACD, BCE, equal to one another, and terminating all the Light which falls upon the Paper from the point where the edges of the Knives meet; eis, fkt, and glv, three hyperbolical Lines representing the Terminus of the Shadow of one of the Knives, the dark Line between the first and second Fringes of that Shadow, and the dark Line between the second and third Fringes of the same Shadow; xip, ykq, and zlr, three other hyperbolical Lines representing the Terminus of the Shadow of the other Knife, the dark Line between the first and second Fringes of that Shadow, and the dark line between the second and third Fringes of the same Shadow. And conceive that these three Hyperbola's are like and equal to the former three, and cross them in the points i, k, and l, and that the Shadows of the Knives are terminated and distinguish'd from the first luminous Fringes by the lines eis and xip, until the meeting and crossing of the Fringes, and then those lines cross the Fringes in the form of dark lines, terminating the first luminous Fringes within side, and distinguishing them from another Light which begins to appear at i, and illuminates all the triangular space ipDEs comprehended by these dark lines, and the right line DE. Of these Hyperbola's one Asymptote is the line DE, and their other Asymptotes are parallel to the lines CA and CB. Let rv represent a line drawn any where upon the Paper parallel to the Asymptote DE, and let this line cross the right lines AC in m, and BC in n, and the six dark hyperbolical lines in p, q, r; s, t, v; and by measuring the distances ps, qt, rv, and thence collecting the lengths of the Ordinates np, nq, nr or ms, mt, mv, and doing this at several distances of the line rv from the Asymptote DD, you may find as many points of these Hyperbola's as you please, and thereby know that these curve lines are Hyperbola's differing little from the conical Hyperbola. And by measuring the lines Ci, Ck, Cl, you may find other points of these Curves.
For instance; when the Knives were distant from the hole in the Window ten Feet, and the Paper from the Knives nine Feet, and the Angle contained by the edges of the Knives to which the Angle ACB is equal, was subtended by a Chord which was to the Radius as 1 to 32, and the distance of the line rv from the Asymptote DE was half an Inch: I measured the lines ps, qt, rv, and found them 0'35, 0'65, 0'98 Inches respectively; and by adding to their halfs the line 1/2 mn, (which here was the 128th part of an Inch, or 0'0078 Inches,) the Sums np, nq, nr, were 0'1828, 0'3328, 0'4978 Inches. I measured also the distances of the brightest parts of the Fringes which run between pq and st, qr and tv, and next beyond r and v, and found them 0'5, 0'8, and 1'17 Inches.