A geometric diagram showing an oval shape on the left connected by converging lines to a circle and point E on the right, with intersecting curves and labeled points S, T, X, and A.

For take any point normal upper C comma m upper D upper C plus n upper C upper B equals thread, and m upper D upper C plus m upper A upper C equals m upper A upper D; therefore take away m upper D upper C, and n upper C upper B tilde m upper A upper C equals thread tilde m upper A upper D equals constant quantity. As m greater than n, normal upper A is called the greater and normal upper B the less focus.

A geometric diagram showing two overlapping circles with a spiral-like curve sweeping outward, connected by lines radiating from a central point A to labeled points around the perimeter.

Proposition 2 is the same as proposition 2 of the Oval.

Proposition 3—Theorem.

If a circle be described with a focus for a center, and the constant difference divided by the power of that focus for a radius, the distance of any point in the curve from the other focus is to the distance from the circle as the power of the central focus is to the power of the other.

Let upper T upper L upper O, upper P upper X upper S be the circles: at any point normal upper C comma upper B upper C colon upper C upper O colon colon m colon n and upper P upper C colon upper C upper A colon colon m colon n.

For m upper C upper A tilde n upper C upper B equals constant difference, but m upper O upper A equals constant difference therefore m upper C upper O equals n upper C upper B and upper B upper C colon upper C upper O colon colon m colon n. QED.

And n upper B upper P equals constant difference therefore m upper C upper A equals n upper C upper P and upper P upper C colon upper C upper A colon colon m colon n. Cor. 1.—If the constant difference equals 0 the curve is a circle. Cor. 2.—If an oval be described with a thread = constant difference, with the same foci and the same powers as the meloid, and any line be drawn from normal upper A, and upper C upper B, upper O upper B, upper V upper B be joined, the angle upper C upper B upper O equals upper V upper B upper O.

For upper B upper C colon upper C upper O colon colon m colon n and upper B upper V colon upper V upper O colon colon m colon n therefore upper B upper C colon upper C upper O colon colon upper B upper V colon upper V upper O therefore upper B upper C colon upper B upper V colon colon upper C upper O colon upper V upper O and upper C upper B upper O equals upper V upper B upper O(6.3).

A two-part geometric diagram: at top, a curved arc with closely clustered points labeled D, E, B, F, and P near its peak, intersected by a dashed vertical line; at bottom, an X formed by two crossing lines with endpoints labeled C, L, D, T, E, and P, intersecting at a central point A or B.

Proposition 4—Theorem.

When the less focus is in the curve, an angle will be formed = that in the Oval (Prop. 4).

For take an indefinitely small arc upper D upper B in the circle, upper C upper B upper D equals upper D upper B upper E (3. Cor. 2), and upper L upper B upper T equals upper T upper B upper P therefore upper C upper B upper L equals upper E upper B upper P.

Or it may be proved as in the oval. If the greater focus normal upper A is at an infinite distance the figure will appear thus:—

A two-part geometric diagram labeled Fig 6 and Fig 7, each showing a curved arc at top with a diamond or kite-shaped figure formed by lines converging to labeled points below, including B, C, K, and A.

Proposition 5—Theorem.

If the distance between the greater focus and the point where the axis cuts the meloid, be to the distance between that point and the less focus, in a greater proportion than the power of the greater focus to that of the less, the curve is convex toward the greater focus at that point, but if the proportion is less, concave.

Let upper A upper D colon upper D upper B colon colon p colon q.

If upper A upper D colon upper D upper B greater than m colon n comma upper C upper D upper E is convex towards normal upper A; but if upper A upper D colon upper D upper B less than m colon n, it is concave towards normal upper A period ellipsis ellipsis Take normal upper C and normal upper E near normal upper D, and upper C upper D equals upper D upper E. Join upper C upper E, upper C upper E cuts the axis in normal upper O. Draw the circle upper C upper L upper E from normal upper B, and upper C upper P upper E from normal upper A. Also draw upper H upper T upper K as in Prop. 3—

Then upper B upper D colon upper D upper T colon colon m colon n colon colon upper B upper C colon upper C upper H, but upper B upper C equals upper B upper L, and upper C upper H equals upper P upper T therefore upper B upper D colon upper D upper T colon colon upper B upper L colon upper P upper T therefore upper B upper D colon upper B upper L colon colon upper D upper T colon upper P upper T therefore upper B upper D colon upper B upper L minus upper B upper D colon colon upper D upper T colon upper P upper T minus upper D upper T therefore upper B upper D colon upper D upper L colon colon upper D upper T colon upper D upper P dot upper B upper D colon upper D upper T colon colon upper D upper L colon upper D upper P therefore upper D upper L colon upper D upper P colon colon m colon n.

As normal upper E and normal upper C are very near normal upper D comma upper A upper D colon upper B upper D colon colon upper A upper C colon upper B upper C, but upper P upper E equals upper L upper E and upper P upper C upper E: 2 right angles colon colon upper P upper E: circumference of upper C upper P upper E, and upper L upper C upper E colon 2 normal upper L colon colon upper L upper E: circumference of upper C upper L upper E, but circ. upper C upper P upper E: circ. upper C upper L upper E colon colon upper A upper C colon upper B upper C colon colon upper A upper D colon upper B upper D colon colon p colon q and upper P upper E equals upper L upper E therefore upper P upper C upper E colon 2 normal upper L colon colon q upper P upper E colon q circ. upper C upper P upper E, and upper L upper C upper E colon 2 normal upper L colon colon p upper P upper E colon p circ. upper C upper L upper E or q circ. upper C upper P upper E therefore upper P upper C upper E colon upper L upper C upper E colon q upper P upper E colon p upper P upper E colon q colon p therefore upper P upper C upper O colon upper L upper C upper O colon q colon p and upper P upper O colon upper O upper L colon colon q colon p; and if p colon q greater than m colon n, upper O upper L colon upper O upper P greater than upper D upper L colon upper D upper P, and normal upper D is nearer to normal upper A than the line upper C upper O upper E, and upper C upper D upper E is convex toward normal upper A; but if p colon q less than m colon n, normal upper D is on the opposite side, and it is concave.

A geometric diagram labeled Fig 7 showing a circle on the left, a larger curved shape in the middle, and an elongated arc on the right, with multiple labeled points connected by a complex web of intersecting lines spanning the figure.

Proposition 6—Problem.

To draw a tangent to a meloid or apioid from a focus without:—

Take m for the power of the greater focus, and n for that of the less, and find the angle upper A upper D upper C (Prop. 5 of the Oval) upon upper A upper B describe a segment upper A upper C upper C upper C upper B containing an equal angle. Join upper B upper C and produce to normal upper H, upper B upper H is a tangent, for suppose normal upper H to be the end of the rod, and take any point normal upper O comma n upper C upper H plus m upper C upper A less than n upper O upper H plus n upper O upper A, therefore normal upper O is without the curve.

upper C upper P is an apioid, upper C upper T is a circle, and upper C upper L is a meloid, with normal upper A and normal upper B as foci.

A geometric diagram showing a long triangular shape converging to a point on the left, connected by intersecting lines to a circle on the right with labeled points along its perimeter.

Proposition 7—Problem.

To draw a tangent to an apioid from any point in the same, the foci and the ratio being given.

It is required to draw a tangent to the apioid at the point bold upper C. Join upper B upper C, and draw a circle as in Prop. 3. Join upper A upper D, and produce it. Make upper A upper P colon upper P upper D colon colon n colon m. Join upper C upper P, and draw upper C upper K at right angles to upper P upper C. Describe a circle through normal upper P, normal upper C, normal upper K, and it was proved in Prop. 7 of the Oval that if any point normal upper O be taken and upper D upper O, upper A upper O joined, upper D upper O colon upper A upper O colon colon m colon n. Suppose normal upper O to be both in the circle and in the apioid, join upper B upper O, then upper L upper O colon upper A upper O colon colon m n, but upper D upper O colon upper A upper O colon m colon n therefore upper L upper O equals upper D upper O, but upper L upper O less than upper D upper O therefore the circle is without the apioid; therefore a tangent normal upper C normal upper T to the circle at normal upper C is a tangent to the apioid.

Axiom 1.

It is possible for a circle to be described touching any given curve internally.

Proposition 8.

To draw a tangent to a meloid at any point normal upper C.

Case 1.—Let the curve be concave towards normal upper B. Describe the circle upper P upper O upper C upper K as in Prop. 7: it will be wholly within the meloid. At normal upper C draw a tangent to the circle: it is also a tangent to the meloid. For let upper R upper N be the tangent to the meloid, it must cut the circle (3.16), and therefore cuts the meloid.

A geometric diagram labeled Fig. 9, Meloid showing a large circle with two smaller overlapping circles inside, containing numerous labeled points connected by an intricate web of intersecting lines.

Case 2.—When the curve is convex towards normal upper B, draw a circle normal upper V as before. Draw upper C upper T a tangent to the circle; it is also a tangent to the meloid. For suppose a circle S drawn touching the curve internally, it must touch normal upper V and also upper C upper T, and any other line would cut normal upper S. QED.

Scholium (see Figs. 8 and 9).—Let normal upper M be the point Join upper A upper M, upper B upper M, cut off upper M upper E, so that upper A upper M colon upper M upper E colon colon power of normal upper B colon power of normal upper A. Join upper A upper E, bisect upper A upper M upper E by upper M upper H, make upper X upper H perpendicular, make upper X upper M upper H equals upper X upper H upper M, upper X upper M is a tangent.

Proposition 9—Theorem.

If lines be drawn from the foci to any point in a meloid or apioid, the sines of the angles which they make with the perpendicular to the tangent are to one another as the powers of the foci see fig. 8, 9.

A page of geometric diagrams labeled Fig 10, Fig 11, and Fig 12, each showing lens-shaped curves formed by converging rays to points labeled A, B, C, D, L, and P, accompanied by handwritten mathematical annotations describing tangent lines, foci, and refraction properties.

the focus See figs 8, 9 For CY is the perpendicular to the tangent and (Prop 8 Oval 6 or 7) Sine ACY : Sine BCY :: (AP : PD) :: m : n

Fig 10 CLD is an oval for converging parallel rays to B, CPD is a oval without AB converging the rays to A.

Fig 11 CLD is an oval for A, B converging rays from A to B CPD is a second for C, B refracting the rays to C.

Fig 12 CHD is a circle not altering rays from A; CTD is a circle as in Prop 8 of the oval refracting the rays as if these had come from B PKLK is a lens of hyperbolas refracting from B to B the whole 3 lenses refract from A to A.