Wikipedia:Reference desk/Archives/Mathematics/2023 February 6

= February 6 =

Help finding references to an old paper
Can anyone help me figure out who the C.J.D. Hill was who sent in some "problems and theorems" to Crelle's Journal, 1832? https://doi.org/10.1515/crll.1832.9.100

Here is a link to a better scan from the internet archive (and the associated diagrams).

Can anyone help me figure out if this particular work was ever referenced by anyone else? Did Crelle's Journal ever print solutions to proposed problems? (It’s sometimes tricky for me to make full sense of these old journals because I don’t read Latin or German and my French is kind of rusty. And for papers this old the metadata indices aren't complete enough to show the citation graph.)

Some of the problems/theorems here involve the spherical geometry analog of the "power of a line" that I was asking about here last month, investigated by Edmond Laguerre in the 1880s. –jacobolus (t) 01:01, 6 February 2023 (UTC)


 * Edit: I looked more carefully, and the “problems and theorems” here were sent in by a variety of authors (Hill only sent in the first few). The one of interest to me is #14 (Fig. 4), sent in by Christoph Gudermann:
 * Google translate + some finessing (so probably not a super accurate translation) renders the problem as:
 * I’m still curious if anyone can find later references to these though. –jacobolus (t) 01:16, 6 February 2023 (UTC)
 * No luck, but since this was posed, without proof, as a problem, this is not entirely unexpected. There is no lack of references to the paper by Möbius a few pages later, in which he presents his inversion formula. BTW, the "small circle" may be any circle on the sphere, also a great circle. --Lambiam 17:11, 6 February 2023 (UTC)
 * It literally says "circle of a sphere", but it must be talking about a small circle. In the great circle case this degenerates, and you can't actually draw such tangent lines (every great circle tangent to a great circle is the circle itself; it's like trying to draw a pencil of lines tangent to a line). You could salvage the quantity of interest (the product of two half-tangents) by only examining the intersection of the two lines, in the same way as you can generalize the power of a point with respect to a circle when the second circle is a point by just taking the squared distance between points. In Laguerre geometry in general, which I have been learning about in the past few weeks and hope to fill in that red link for, the fundamental objects are (oriented) lines (on the sphere, oriented great circles), and the derived objects (from envelopes of lines) are circles, points, etc. –jacobolus (t) 17:35, 6 February 2023 (UTC)
 * It is degenerate in the sense that the pencil is very narrow, since all tangents coincide, but they still exist and intersect the original great circle – in two points, except for the even more degenerate case that both circles coincide. But pick two points of intersection, and the angles are defined. --Lambiam 22:25, 6 February 2023 (UTC)
 * To partially answer my own question, Gudermann followed up by including this (or something related anyway) in his 1835 book Lehrbuch der niederen Sphärik §296–297. (If there’s one thing OCR loves, it’s a mediocre scan of a book set entirely in a Fraktur font.) –jacobolus (t) 17:18, 6 February 2023 (UTC)
 * {{color|#094| Kufgabe« 9)tan foQ ben Bufammeniftans unter ben SBinfeln
 * ftnben, tvel^^e }n>e{ Zongenten dneft Areifeö mit einem burdj^ i^ren
 * 64^eitel ge^enben <^au)>tfreife mad)ttL
 * Slttflifung. 3n Fig. 166 feien PA unb PB gwei Spangen*
 * ten eineft JtreifeS, unb burc^ ben Sd^eitef P il^reS SBinfelS gefie ber
 * «^uptbogen XY im inneren be« Sßinteld APB. ^an üt^t n^
 * bie Siabien MA, MB, femer na^ bem @di^eitel beS mnUti P bie
 * Sinie MP unb auf XY baS fot^ Mm ; bann i^ im 2>reie(!e PMA }} etc. jacobolus (t) 17:27, 6 February 2023 (UTC)
 * Try newocr.com; it has an option for German Fraktur. Not perfect, but much better:
 * Aufgabe. Man soll den Zusammenhang unter den Winkeln finden, welche zwei Tangenten eines Greis es rnit einein durch ihren Scheitel gehenden Hauptlreise machen. . Au flösung. In Fig. 166 seien PA und PB zwei Zangenten eines Kreises, und durch den Scheitel P ihres Winkels gehe der Hauptdogen xY inr Inneren des Winkels APR Man siehe noch die Radien MA, ME, ferner nach dem Scheitel des Win els P die Linie Mk und auf xY das Loth Mmz dann ist im Dreiecke PMA ...
 * --Lambiam 22:07, 6 February 2023 (UTC)
 * Now I just need a psychic to figure out what the figures showed – whatever temp worker the Google book scan department hired scanned all of the pages of figures folded up. :-)
 * Anyone know if there are other scans of Lehrbuch der niederen Sphärik online, ideally with the figures scanned too? –jacobolus (t) 00:57, 7 February 2023 (UTC)
 * Power of a line Gudermann 1835 Fig. 166.pnge 166 is fully visible, while the hidden Figure 167 is less essential. --Lambiam 12:47, 7 February 2023 (UTC)
 * Cleaned up first page: [... moved to subsection ...]
 * Gotta do some errands before proceeding with the rest. –jacobolus (t) 20:44, 7 February 2023 (UTC)
 * I made a few minor corrections. --Lambiam 23:27, 7 February 2023 (UTC)
 * Next part: [...] –jacobolus (t) 01:46, 9 February 2023 (UTC)
 * A few more minor corrections. --Lambiam 12:02, 8 February 2023 (UTC)
 * One notable thing here is that Gudermann was not operating from the concept of directed lines/circles, so from the later Laguerre perspective he was picking one of the tangent lines to be the wrong orientation (which is why this version is a quotient rather than a product). The version sent in to Crelle's "correctly" used two tangents oriented the same way with respect to the circle. –jacobolus (t) 23:52, 7 February 2023 (UTC)
 * A few more minor corrections. --Lambiam 12:02, 8 February 2023 (UTC)
 * One notable thing here is that Gudermann was not operating from the concept of directed lines/circles, so from the later Laguerre perspective he was picking one of the tangent lines to be the wrong orientation (which is why this version is a quotient rather than a product). The version sent in to Crelle's "correctly" used two tangents oriented the same way with respect to the circle. –jacobolus (t) 23:52, 7 February 2023 (UTC)

CJD Hill
Not sure if it's the right person, but if you're still interested in the identity of CJD Hill, the Mathematics Genealogy Project lists one Carl Johan Danielsson Hill who received his PhD from Lund University in 1817. GalacticShoe (talk) 16:49, 6 February 2023 (UTC)
 * Thanks. I had figured him out before; this guy: sv:Carl Johan Hill. But I don't care about Hill after all. :-) –jacobolus (t) 17:08, 6 February 2023 (UTC)
 * Perhaps his discovery that $$x(a\oplus b)=x(a)\oplus x(b),$$ where $$x(a)$$ denotes "the roots" of $$x^3-3ax^2-3x+a=0,$$ ties this in with your interests. (I'm not quite sure how to interpret the plural; this may simply hold for each of the three branches, which are separated by the roots of $$3x^2-1.$$) --Lambiam 10:47, 7 February 2023 (UTC)
 * Fair enough. I did indeed notice that bit. I’m not quite sure how to make sense of it though. :-) –jacobolus (t) 11:54, 7 February 2023 (UTC)
 * So out of curiosity I played around a little with the formula. For convenience of any readers out there, $$p \oplus q$$ here is defined as $$\frac{p+q}{1-pq}$$, and I'm going to write $$P(a)$$ for $$x^{3} - 3ax^{2} - 3x + a$$. As it turns out, what Hill means is that, for fixed $$a, b$$, every possible pair of roots $$x(a), x(b)$$ of $$P(a), P(b)$$ respectively has the property that $$x(a) \oplus x(b)$$ is a root of $$P(a \oplus b)$$. In fact, if we fix a root $$x(a)$$, then varying $$x(b)$$ across the three possible roots of $$P(b)$$ goes through all three roots of $$P(a \oplus b)$$. Same for fixing $$x(b)$$ and varying $$x(a)$$. GalacticShoe (talk) 13:04, 12 February 2023 (UTC)
 * I should also note for the preceding message that the fact about fixing one root and varying the other is not particularly surprising, since when $$x$$ or $$y$$ is fixed, $$\frac{x+y}{1-xy}$$ is injective. GalacticShoe (talk) 13:11, 12 February 2023 (UTC)
 * Thanks for making that explicit. To be clear: I understood what Hill’s statement literally means. While I didn’t hunt down his proof, try to prove it myself, or do any numerical experiments, I am willing to believe that it is a true statement.
 * When I say “I’m not quite sure how to make sense of it” what I mean is: I haven’t figured out why (at a conceptual level) the specific form of the polynomial $$x^3 - 3ax^2 - 3x + a = 0$$ leads to this property, whether there’s a convenient way to express $$x^3 - 3ax^2 - 3x + a$$ in terms of $$\oplus$$ that would make it clear why this happens, whether there are polynomials of other degree with a similar property, and so on. I expect it would take some considerable research effort (for me anyway; perhaps some experts already know about this or could grasp it immediately) to investigate what is going on here. But it does seem curious and I would be interested to hear any insights. –jacobolus (t) 18:44, 12 February 2023 (UTC)
 * To start, given roots $$\alpha_1, \alpha_2, \alpha_3,$$ of $$P(a)$$ we should be able to factor $$(x - \alpha_1)(x - \alpha_2)(x - \alpha_3) = x^3 - 3ax^2 - 3x + a.$$ So if we also have $$\beta_1, \beta_2, \beta_3,$$ of $$P(b)$$ and $$\gamma_1, \gamma_2, \gamma_3,$$ of $$P(c) = P(a \oplus b),$$ then Hill's claim is $$\alpha_i \oplus \beta_j = \gamma_k$$ for some $$i,j \in \{1,2,3\}$$ given any $$k \in \{1,2,3\}.$$ We might start from:
 * $$\begin{align}

3a &= \alpha_1 + \alpha_2 + \alpha_3, \\ -3 &= \alpha_1\alpha_2 + \alpha_2\alpha_3 + \alpha_3\alpha_1 \\ -a &= \alpha_1\alpha_2\alpha_3 \\[20mu] 3b &= \beta_1 + \beta_2 + \beta_3, \\ -3 &= \beta_1\beta_2 + \beta_2\beta_3 + \beta_3\beta_1 \\ -b &= \beta_1\beta_2\beta_3 \\[20mu] 3(a\oplus b) &= \gamma_1 + \gamma_2 + \gamma_3, \\ -3 &= \gamma_1\gamma_2 + \gamma_2\gamma_3 + \gamma_3\gamma_1 \\ -(a\oplus b) &= \gamma_1\gamma_2\gamma_3 \end{align}$$
 * But I think I’d need to fiddle with this for a good while to figure it out. –jacobolus (t) 19:01, 12 February 2023 (UTC)
 * One more note in case it helps anyone. Being roots of the polynomial $$P$$ implies the relations:
 * $$\begin{align}

\alpha_1 \oplus \alpha_2 \oplus \alpha_3 &= \frac{\alpha_1 + \alpha_2 + \alpha_3 - \alpha_1\alpha_2\alpha_3}{1 - \alpha_1\alpha_2 - \alpha_2\alpha_3 - \alpha_3\alpha_1} = a \\ \beta_1 \oplus \beta_2 \oplus \beta_3 &= \frac{\beta_1 + \beta_2 + \beta_3 - \beta_1\beta_2\beta_3}{1 - \beta_1\beta_2 - \beta_2\beta_3 - \beta_3\beta_1} = b \\ \gamma_1 \oplus \gamma_2 \oplus \gamma_3 &= \frac{\gamma_1 + \gamma_2 + \gamma_3 - \gamma_1\gamma_2\gamma_3}{1 - \gamma_1\gamma_2 - \gamma_2\gamma_3 - \gamma_3\gamma_1} = a \oplus b \end{align}$$
 * So at the very least it is immediately clear why:
 * $$\gamma_1 \oplus \gamma_2 \oplus \gamma_3 = \alpha_1 \oplus \alpha_2 \oplus \alpha_3 \oplus \beta_1 \oplus \beta_2 \oplus \beta_3.$$
 * –jacobolus (t) 19:11, 12 February 2023 (UTC)
 * Okay, I figured out what is going on and now I feel a bit bamboozled. Our polynomial equation can be written as $$x^{\oplus3} := x\oplus x \oplus x = a,$$ which has roots $$x = a^{\oplus1/3}, a^{\oplus1/3} \oplus {\sqrt3}, a^{\oplus1/3} \ominus {\sqrt3}.$$ In other words, the three roots of this equation are equally spaced around the circle, treating the line as a model for the circle under the relation $$\theta = 2\arctan{x}$$ or $$z = (1 + xi)\big/(1 - xi).$$ That is, this the equation $$z^3 = z_a,$$ with the complex unit circle projected to the real line. –jacobolus (t) 21:29, 12 February 2023 (UTC)

OCR / Cleanup of Lehrbuch der niederen Sphärik



 * Aufgabe. Man soll den Zusammenhang unter den Winkeln finden, welche zwei Tangenten eines Kreises mit einem durch ihren Scheitel gehenden Hauptkreise machen.
 * Auflösung. In Fig. 166 seien $$PA$$ und $$PB$$ zwei Tangenten eines Kreises, und durch den Scheitel $$P$$ ihres Winkels gehe der Hauptbogen $$XY$$ im Inneren des Winkels $$APB.$$ Man ziehe noch die Radien $$MA, MB,$$ ferner nach dem Scheitel des Winkels $$P$$ die Linie $$MP$$ und auf $$XY$$ das Loth $Mm$; dann ist im Dreiecke $$PMA$$
 * $$\sin MA = \sin PM \cdot \sin APM,$$
 * und im Dreiecke $$PMm$$ ist $$\sin Mm = \sin PM \cdot \sin mPM$$; daher hat man
 * $$\frac{\sin MA}{\sin Mm} = \frac{\sin APM}{\sin mPM}.$$
 * Bezeichnet man nun den Radius des Kreises mit $$r$$ und das Loth $$Mm$$ mit $$u,$$ so ist also
 * $$\frac{\sin r}{\sin u} = \frac{\sin \tfrac12 APB}{\sin mPM}.$$
 * Bezeichnet man ferner die Winkel $$APY$$ und $$BPY$$ mit $$\alpha$$ und $$\beta,$$ so ist $$APB = \alpha + \beta,$$ also $$MPA = \tfrac12(\alpha + \beta),$$ daher ist $$mPM = \tfrac12(\alpha + \beta) - \alpha = \tfrac12(\beta - \alpha),$$ folglich hat man die Gleichung
 * 1. $$\frac{\sin u}{\sin r} = \frac{\sin \tfrac12(\beta - \alpha)}{\sin \tfrac12(\beta + \alpha)}.$$


 * Erhebt man die Gleichung zum Quadrate und subtrahirt man sie aus beiden Seiten von Eins, so hat man

\frac{\sin r^2 - \sin u^2}{\sin r^2} = \frac{\sin\tfrac12(\beta + \alpha)^2 - \sin\tfrac12(\beta - \alpha)^2}{\sin\tfrac12(\beta + \alpha)^2} $$ d.h.
 * 2. $$

\frac{\sin mn \cdot \sin mn' }{\sin r^2} = \frac{\sin \alpha \cdot \sin \beta }{\sin \tfrac12(\beta + \alpha)^2},$$ oder

\frac{\sin mn \cdot \sin mn'}{ \sin \tfrac12 nn'^2 } = \frac{\sin APY \cdot \sin BPY}{\sin \tfrac12APB^2}.$$
 * Ferner folgt aus der Gleichung (1)

\frac{\sin r - \sin u}{\sin r + \sin u } = \frac{\sin\tfrac12(\beta + \alpha) - \sin\tfrac12(\beta - \alpha) } {\sin\tfrac12(\beta + \alpha) + \sin\tfrac12(\beta - \alpha) } $$ oder
 * 3. $$\frac{\operatorname{tng}\tfrac12(r - u)}{\operatorname{tng}\tfrac12(r + u)} = \frac{\operatorname{tng}\tfrac12\alpha}{\operatorname{tng}\tfrac12\beta}.$$
 * Befindet sich die Linie $$XY$$ außerhalb des Winkels, durch dessen Scheitel sie geht, und also auch außerhalb des Kreises, wie in Fig. 167, so gelten die vorigen Formeln ebenfalls, und die Herleitung für diesen Fall stimmt mit der vorigen überein.
 * Anmerkung. Ein Winkel, in dessen Innerem sich ein Kreis befindet, und dessen Schenkel Tangenten dieses Kreises sind, heiße ein um den Kreis geschriebener Winkel.


 * Lehrsatz. Sind zwei Winkel um denselben Kreis geschrieben, und zieht man durch ihre Scheitel einen Hauptbogen, so theilt dieser jeden Winkel (innerlich oder äußerlich) in zwei Theile, so, daß erstens die Sinus der halben Unterschiede der Theile dieser Winkel sich zu einander verhalten, wie die Sinus der halben Winkel selbst; zweitens die Producte aus den Sinus der Theile der Winkel sich zu einander verhalten, wie die Quadrate der Sinus der halben Winkel; endlich drittens, daß die Tangenten der halben Theile des einen Winkels proportional sind zu den Tangenten der halben Theile des anderen Winkels.
 * Beweis. Zieht man in Fig. 166 und Fig. 167 von einem anderen Punkte $$P'$$ im Hauptbogen $$XY$$ die Tangenten $$P'A'$$ und $$P'B',$$ und setzt man den Winkel $$A'P'Y = \alpha',$$ $$B'P'Y = \beta',$$ so ist, wenn die übrige Bezeichnung des §. 296 beibehalten wird,

\frac{\sin u}{\sin r} = \frac{\sin\tfrac12(\beta - \alpha)} {\sin\tfrac12(\beta + \alpha)} $$ und $$ \frac{\sin u}{\sin r} = \frac{\sin\tfrac12(\beta' - \alpha')} {\sin\tfrac12(\beta' + \alpha')}, $$ also

\frac{\sin\tfrac12(\beta - \alpha)} {\sin\tfrac12(\beta + \alpha)} = \frac{\sin\tfrac12(\beta' - \alpha')} {\sin\tfrac12(\beta' + \alpha')}, $$
 * Ferner ist $$

\frac{\sin mn \cdot \sin mn'}{\sin r^2} = \frac{\sin APY \cdot \sin BPY}{\sin\tfrac12 APB^2} $$ und auch

\frac{\sin mn \cdot \sin mn'}{\sin r^2} = \frac{\sin A'PY \cdot \sin B'P'Y}{\sin \tfrac12 A'P'B'^2},$$ also

\frac{\sin APY \cdot \sin BPY}{\sin\tfrac12 APB^2} = \frac{\sin A'P'Y \cdot \sin B'P'Y}{\sin \tfrac12 A'P'B'^2}.$$
 * Endlich ist $$

\frac{\operatorname{tng}\tfrac12 \alpha} {\operatorname{tng}\tfrac12 \beta} = \frac{\pm\operatorname{tng}\tfrac12 (r-u)} {\operatorname{tng}\tfrac12 (r+u)}$$ und auch

\frac{\operatorname{tng}\tfrac12 \alpha'} {\operatorname{tng}\tfrac12 \beta'} = \frac{\pm\operatorname{tng}\tfrac12 (r-u)} {\operatorname{tng}\tfrac12 (r+u)},$$ und also

\frac{\operatorname{tng}\tfrac12 \alpha} {\operatorname{tng}\tfrac12 \beta} = \frac{\operatorname{tng}\tfrac12 \alpha'} {\operatorname{tng}\tfrac12 \beta'}.$$
 * Zusatz. Zieht man vom Punkte $$m$$ aus die Tangenten $$m\mu$$ und $$n\nu,$$ so machen sie mit $$XY$$ die Winkel $$\nu mY = \beta$$ und $$\mu mY = \alpha,$$ und da nun $$\alpha + \beta = 180^\circ$$ also $$\tfrac12(\alpha + \beta) = 90^\circ,$$ ist, so ist $$\beta = 180^\circ - \alpha,$$ also $$\tfrac12(\beta - \alpha) = 90^\circ - \alpha''$$; und also
 * 1. $$\cos \alpha'' = \frac{\sin\tfrac12(\beta - \alpha)}{\sin\tfrac12(\beta + \alpha)},$$
 * 2. $$\frac{\sin\alpha \cdot \sin\beta}{\sin \tfrac12(\beta - \alpha)} = \frac{\sin\alpha^2}{\cos\alpha^2} = \operatorname{tng}\alpha''^2,$$
 * 3. $$\frac{\operatorname{tng}\tfrac12\alpha}{\operatorname{tng}\tfrac12\beta} = \operatorname{tng}\tfrac12\alpha''^2.$$
 * Anmerkung 1. Die vorstehenden Sätze sind nur verschiedene Formen des Ausdrucks eines einzigen, welcher der reciproke von dem im §. 255 bewiesenen Satze ist, und auch daraus auf eine einfache Art durch Construction des reciproken Kreises hätte hergeleitet werden können. Die beiden Sätze erscheinen noch gleichlautender, wenn man sich nicht die um den Kreis geschriebenen Winkel $$APB$$ und $$A'P'B',$$ sondern ihre Nebenwinkel von der Linie $$XY$$ getheilt vorstellt. Die zweiten Schenkel dieser Nebenwinkel sind dann Tangenten eines anderen Kreises, welcher der Gegenkreis des ersten ist.
 * Anmerkung 2. Die hier behandelten Formeln gelten auch unverändert im analogen Falle der Planimetrie, und sind auch in dieser Hinsicht noch neu. Sie drücken auch dort das Reciproke von dem Gesetze aus, nach welchen die aus den Theilen zweier sich schneidenden Sehnen eines Kreises construirten Rechtecke gleich groß sind.


 * [...]
 * Will man die Methode der reciproken Polaren anwenden so steht auch sie der Sphärik zu Gebote, und da ihre Anwendung in der Sphärik nicht verwickelter als in der Planimetrie ist, so sieht man also, daß in der Sphärik eine zweifache Methode der Reciprocität anwendbar ist, wovon aber derjenigen, welche oben gebraucht worden ist, unstreitig ein großer Vorzug vor der anderen gebührt; nur gibt es zu ihr kein Analogon in der Planimetrie, oder man müßte sich alle geraden Linien als Hauptbogen auf einer Kugel vorstellen. Aber dieser Reichthum der Sphärik an Methoden macht sie lehrreicher, interessanter und überhaupt wichtiger als die Planimetrie, deren weitere Ausbildung durch die der Sphärik bedingt wird, wie das in der Anmerkung 2. zu §. 297 erwähnte Beispiel, und eine Menge anderer, welche zum Theil noch interessanter sein dürften, zur Genüge zeigen.

Feel free to make further corrections. Maybe folks who read German can also help clean up translation errors if I paste a machine-translated version after this. –jacobolus (t) 23:47, 8 February 2023 (UTC)

Machine translation massaged to hopefully be more or less mathematically correct.




 * Problem. Find the relation between the angles which two tangents of a circle make with a great circle passing through their intersection.
 * Solution. Let $$PA$$ and $$PB$$ in Fig. 166 be two tangents of a circle, with the great-circle arc $$XY$$ passing through the vertex $$P$$ inside their angle $$APB.$$ Draw the radii $$MA, MB,$$ and also the line $$MP$$ through the vertex of the angle $$P$$ and from $$XY$$ the altitude $Mm$; then in the triangle $$PMA$$ we have
 * $$\sin MA = \sin PM \cdot \sin APM,$$
 * and in the triangle $$PMm$$ we have $$\sin Mm = \sin PM \cdot \sin mPM$$; therefore we have
 * $$\frac{\sin MA}{\sin Mm} = \frac{\sin APM}{\sin mPM}.$$
 * Denoting the radius of the circle by $$r$$ and the altitude $$Mm$$ by $$u,$$ we thus have
 * $$\frac{\sin r}{\sin u} = \frac{\sin \tfrac12 APB}{\sin mPM}.$$
 * If we further denote the angles $$APY$$ and $$BPY$$ by $$\alpha$$ and $$\beta,$$ then $$APB = \alpha + \beta,$$ and thus $$MPA = \tfrac12(\alpha + \beta),$$ therefore $$mPM = \tfrac12(\alpha + \beta) - \alpha = \tfrac12(\beta - \alpha),$$ hence we have the equation
 * 1. $$\frac{\sin u}{\sin r} = \frac{\sin \tfrac12(\beta - \alpha)}{\sin \tfrac12(\beta + \alpha)}.$$


 * Squaring this equation and subtracting each side from unity, we have

\frac{\sin r^2 - \sin u^2}{\sin r^2} = \frac{\sin\tfrac12(\beta + \alpha)^2 - \sin\tfrac12(\beta - \alpha)^2}{\sin \tfrac12(\beta + \alpha)^2} $$ i.e.
 * 2. $$

\frac{\sin mn \cdot \sin mn' }{\sin r^2} = \frac{\sin \alpha \cdot \sin \beta }{\sin \tfrac12(\beta + \alpha)^2},$$ or

\frac{\sin mn \cdot \sin mn'}{ \sin \tfrac12 nn'^2 } = \frac{\sin APY \cdot \sin BPY}{\sin \tfrac12APB^2}.$$
 * Furthermore, it follows from equation (1) that

\frac{\sin r - \sin u}{\sin r + \sin u } = \frac{\sin\tfrac12(\beta + \alpha) - \sin\tfrac12(\beta - \alpha) } {\sin\tfrac12(\beta + \alpha) + \sin\tfrac12(\beta - \alpha) } $$ or
 * 3. $$\frac{\operatorname{tng}\tfrac12(r - u)}{\operatorname{tng}\tfrac12(r + u)} = \frac{\operatorname{tng}\tfrac12\alpha}{\operatorname {tng}\tfrac12\beta}.$$
 * If the line $$XY$$ is outside the angle through whose vertex it passes, and therefore also outside the circle, as in Fig. 167, the previous formulas apply equally, and the derivation for this case is the same as the previous one.
 * Note. An angle inside which there is a circle and whose legs are tangents to this circle is called a circumscribed angle of the circle.


 * Theorem. If two angles are circumscribed angles of the same circle, and a great-circle arc is drawn through their vertices, this divides each angle (internal or external) into two parts, so that, first, the sines of the half differences of the parts of these angles are related to one another as the sines of the half-angles themselves; secondly, the products of the sines of the parts of the angles are related to each other as the squares of the sines of the half angles; finally, thirdly, that the tangents of halve one angle are proportional to the tangents of halve the other angle.
 * Proof. If we draw, in Fig. 166 and Fig. 167, the tangents $$P'A'$$ and $$P'B'$$ from another point $$P'$$ on the great-circle $$XY,$$ and set the angle $$A'P'Y = \alpha',$$ $$B'P'Y = \beta',$$ then we have, keeping the other notations from § 296 unchanged,

\frac{\sin u}{\sin r} = \frac{\sin\tfrac12(\beta - \alpha)} {\sin\tfrac12(\beta + \alpha)} $$ and $$ \frac{\sin u}{\sin r} = \frac{\sin\tfrac12(\beta' - \alpha')} {\sin\tfrac12(\beta' + \alpha')}, $$ so

\frac{\sin\tfrac12(\beta - \alpha)} {\sin\tfrac12(\beta + \alpha)} = \frac{\sin\tfrac12(\beta' - \alpha')} {\sin\tfrac12(\beta' + \alpha')}. $$
 * Furthermore, $$

\frac{\sin mn \cdot \sin mn'}{\sin r^2} = \frac{\sin APY \cdot \sin BPY}{\sin \tfrac12 APB^2} $$ and also

\frac{\sin mn \cdot \sin mn'}{\sin r^2} = \frac{\sin A'PY \cdot \sin B'P'Y}{\sin \tfrac12 A'P'B'^2},$$ so

\frac{\sin APY \cdot \sin BPY}{\sin \tfrac12 APB^2} = \frac{\sin A'P'Y \cdot \sin B'P'Y}{\sin \tfrac12 A'P'B'^2}.$$
 * Finally, $$

\frac{\operatorname{tng}\tfrac12 \alpha } {\operatorname{tng}\tfrac12 \beta} = \frac{\pm\operatorname{tng}\tfrac12 (r-u)} {\operatorname{tng}\tfrac12 (r+u)}$$ and also

\frac{\operatorname{tng}\tfrac12 \alpha'} {\operatorname{tng}\tfrac12 \beta'} = \frac{\pm\operatorname{tng}\tfrac12 (r-u)} {\operatorname{tng}\tfrac12 (r+u)},$$ and so

\frac{\operatorname{tng}\tfrac12 \alpha } {\operatorname{tng}\tfrac12 \beta} = \frac{\operatorname{tng}\tfrac12 \alpha'} {\operatorname{tng}\tfrac12 \beta'}.$$
 * Corollary. If one draws the tangents $$m\mu$$ and $$n\nu$$ from the point $$m$$, they form with $$XY$$ the angles $$\nu mY = \beta$$ and $$\mu mY = \alpha,$$ and since $$\alpha + \beta = 180 ^\circ$$ and so $$\tfrac12(\alpha + \beta) = 90^\circ,$$ therefore $$\beta = 180^\circ - \alpha,$$ and so $$\tfrac12(\beta - \alpha) = 90^\circ - \alpha''$$; and thus
 * 1. $$\cos \alpha'' = \frac{\sin\tfrac12(\beta - \alpha)}{\sin \tfrac12(\beta + \alpha)},$$
 * 2. $$\frac{\sin\alpha \cdot \sin\beta }{\sin \tfrac12(\beta - \alpha)} = \frac{\sin \alpha^2}{\cos \alpha ^2} = \operatorname{tng}\alpha''^2,$$
 * 3. $$\frac{\operatorname{tng}\tfrac12\alpha}{\operatorname{tng}\tfrac12\beta} = \operatorname{tng}\tfrac12\alpha''^2.$$
 * Note 1. The above propositions are only different ways of expressing a single one, which is the dual of the proposition proved in § 255, and could also have been derived from it in a simple way by constructing the dual circle. The two propositions appear even more identical if one does not look at the angles $$APB$$ and $$A'P'B',$$ circumscribed around the circle, but at their secondary angles divided by the line $$XY.$$ The second legs of these secondary angles are then tangents of another circle, which is the opposite circle of the first.
 * Note 2. The formulas dealt with here also apply without change in the analogous case of planimetry, and are also new in that context. There they also express the dual of the law according to which the product of each pair of segments from two intersecting chords of a circle are of equal magnitude.

Feel free to fix any mistakes or change this translation to make it clearer or more faithful to the original if it doesn't sacrifice comprehensibility to a modern English-reading audience. –jacobolus (t) 02:16, 10 February 2023 (UTC)