# 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.

**Author(s):** Edward Waring  
**Year:** 1765  
**Journal:** Philosophical Transactions (1683-1775)  
**Volume:** 55  
**Pages:** 5 pages  
**Identifier:** jstor-105457  
**JSTOR URL:** <https://www.jstor.org/stable/105457>  

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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.