Two Theorems, by Edward Waring, M. A. Lucasian Professor of Mathematics in the University of Cambridge, and F. R. S. In a Letter to Charles Morton, M. D. Sec. R. S.

XXII. Two Theorems, by Edward Waring, M. A. Lucasian Professor of Mathematics in the University of Cambridge, and F. R. S. In a Letter to Charles Morton, M. D. Sec. R. S. THEOREMA I. FIGURA I. Read April 25, 1765. IN datâ Ellipsi inscribantur duo (n) Laterum Polygona \(abcde\), &c. et \(pqrst\), &c. ad Puncta respectiva \(a, b, c, d, e, \&c.\). \(p, q, r, s, t, \&c.\) ducantur Tangentes \(AB, BC, CD, DE, \&c.\) et \(PQ, QR, RS, ST, \&c.\) et sint \[\angle abB = \angle cbC, \quad \angle bcC = \angle dcD, \quad \angle cdD = \angle edE, \&c.\] et \[\angle pqQ = \angle rqR, \quad \angle qrR = \angle srS, \quad \angle rsS = \angle tsT,\] et sic deinceps. Et erit Summa Laterum \[ab + bc + cd + de + \&c. = pq + qr + rs + st + \&c.\] FIGURA 2. Cor. Ducatur in Ellipsi Polygonum \(abcde\) &c. (n) Laterum Methodo supra traditâ; inscribatur etiam aliud Polygonum \(abhklm\) &c. (n) Laterum quovis alio. alio Modo, cujus unus Angulus ponitur ad Punctum (a), et Summa $ab + bc + cd + de + \ldots$ &c. major est quam Summa $ab + bk + kl + lm + \ldots$ &c. **THEOREMA II.** **TAB. IV. FIGURA I.** Describantur circa datam Ellipsim duo (n) Laterum Polygona ABCDE &c. et PQRS T &c. quorum Puncta Contactuum respective sunt $a, b, c, d, e,$ &c. et $p, q, r, s, t,$ &c. Et sint $\text{Tang.} + \text{Seca. Comp.} \angle aBb : \text{Tan.} + \text{Seca. Comp.}$ $\angle cCb :: bC : bB,$ et $\text{Tang.} + \text{Seca. Comp.} \angle cCb : \text{Tan.} + \text{Seca. Comp.}$ $\angle cDd :: cD : cC,$ et $\text{Tang.} + \text{Seca. Comp.} \angle cDd : \text{Tan.} + \text{Seca. Comp.}$ $\angle eEd :: Ed : aD$ &c. Et sic $\text{Tang.} + \text{Seca. Comp.} \angle pQq : \text{Tan.} + \text{Seca. Comp.}$ $\angle qRr :: qR : qQ,$ et $\text{Tang.} + \text{Seca. Comp.} \angle qRr : \text{Tan.} + \text{Seca. Comp.}$ $\angle sSr :: Sr : rR,$ et $\text{Tang.} + \text{Seca. Comp.} \angle sSr : \text{Tan.} + \text{Seca. Comp.}$ $\angle tTs :: Ts : sS,$ et sic deinceps. Et erit Summa Laterum $AB + BC + CD + DE + \ldots = PQ + QR + RS + ST + \ldots$ **FIGURA** Figura 3. Cor. Describatur circa Ellipsim Polygonum (n) Laterum A B C D E, &c. Methodo, quae prius data fuit; Describatur etiam circa Ellipsim aliud Polygonum G H K L M, &c. (n) Laterum quavis aliâ Methodo, cujus unum Punctum Contactus (a) est Punctum Contactus Polygoni A B C D E, &c. Et Summa A B + B C + C D + D E + &c. minor erit quam Summa G H + H K + K L + L M + &c. Consimiles Proprietates affirmari possunt de Polygonis Hyperbolas descriptis, &c.