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THE AMERICAN MATHEMATICAL MONTHLY, Founded

IN 1894 BY BENJAMIN F. FINKEL, WAS PUBLISHED BY HIM UNTIL 1913. FROM 1913 TO 1916 IT WAS OWNED AND PUBLISHED

BY REPRESENTATIVES OF FOURTEEN UNIVERSITIES AND COLLEGES

IN THE MIDDLE WEST

VOLUME XXVII

1920

PUBLISHED BY THE ASSOCIATION

LANCASTER, PA., AND PROVIDENCE, R. I.

1

PRESS OF
HE NEW ERA PRINTING COMPANY
LANCASTER, PA.

280248

THE AMERICAN

MATHEMATICAL MONTHLY

The endless torch race first began

Who knoweth where? Who knoweth when?

The runners give from hand to hand,—
In lines of flame from land to land;

And still again, and yet again,

They follow straight the soul's command,-
Though but in part they understand,—
And still pass on the deathless brand

From man to man-and we are men!

WILLIAM ADAMS SLADE.

THE APRIL MEETING OF THE IOWA SECTION.

The Iowa Section of the Mathematical Association of America met for its fourth regular meeting April 26, 1919, in connection with the Iowa Academy of Science, at the Iowa State Teachers College, Cedar Falls, Iowa. There were present as members of the Association: L. M. Coffin, Coe College; I. S. Condit, Iowa State Teachers College; F. M. McGaw, Cornell College; J. F. Reilly, Uni'versity of Iowa; H. L. Rietz, University of Iowa; Maria M. Roberts, Iowa State College; F. M. Weida, University of Iowa; and C. W. Wester, Iowa State Teachers College.

The following papers were read and discussed:

"The teaching of a first course in mathematics." By Professor H. L. Rietz. "A course in arithmetic." By Professor I. S. Condit.

"Effect of delaying algebra until the tenth grade." By Miss Lida Pittman, Fort Dodge (by invitation).

"An example of curve fitting." By G. W. Snedecor, Ames (by invitation). "Outlines of a course in trigonometry." By Professor J. F. Reilly. "Some analogies between algebraic equations and linear differential equations." By Peter Luteyn, State Teachers College (by invitation).

The following were elected officers for the ensuing year: Chairman, I. S. CONDIT; Vice Chairman, F. M. MCGAW; Secretary-Treasurer, L. M. COFFIN.

C. W. WESTER, Secretary-Treasurer.

SOME EXTENSIONS OF THE WORK OF PAPPUS AND STEINER ON TANGENT CIRCLES.

By J. H. WEAVER, Ohio State University.

Introduction. The figure of three mutually tangent semicircles with their centers in the same straight line was known among the Greeks as the "Shoemaker's Knife" (äpßnλos). A few of the properties of the figure are found in the works of Archimedes.1 Others occur in the Collection of Pappus.2 After the Greek period we find no work done on the problem until Steiner generalized the results of Pappus and added several others dealing with the perspective properties of the figure. Later Sir Thomas Muir added a theorem giving formulæ for various sets of radii involved. Habicht has discussed some of the properties of elliptic functions connected with the figure while M. G. Fontené has generalized certain formulæ arising from sets of tangent circles.

In the present paper formulæ for the radii of certain sets of circles are developed and used to build up several types of infinite series which may be summed geometrically. Then some general properties of tangents and normals to conics

S2

S1

A

1 02

03

FIG. 1.

associated with three mutually tangent circles are set forth. These properties lead to a quadrangular-quadrilateral configuration and incidentally furnish some methods for con structing conics. And finally some theorems connected with centers of perspectivity of the various sets of circles are proved.

1. General Considerations. Let there be two circles tangent internally, with centers O1 and O2 and let a circle with center Sn (n = 1, 2, ...) (Fig. 1) be tangent to these. Then S, lies on an ellipse with foci O1 and O2.

If we take the midpoint of 0102 as origin and 0102 as the x-axis, the equation of the ellipse will be

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where r1 and r2 (r1⁄2 > r1) are the radii of the circles (01) and (02) respectively.

1 Works of Archimedes, ed. Heath, Cambridge, 1897, Lemmas, 4-6.

2 Collectio, ed. Hultsch., Berlin, 1876-8, Vol. I, pp. 209 and ff.

3 Steiner, Gesammelte Werke, Berlin, 1881, Vol. 1, pp. 47-76.

4 Proceedings of Edinburgh Math. Soc., Vol. 3, p. 119. In the same volume, pages 2-11, J. S. Mackay has collected some of the simpler theorems connected with the problem.

5 Konrad Habicht, Die Steinerschen Kreisreihen, Berne, 1904, 35 pp. In this work are found extensive references bearing on the subject.

"Sur les cercles de Pappus," Nouvelles Annales de Mathématiques (4), tome 1918, pp. 383–90. 7 The center of a circle tangent to two given circles lies on a conic having the centers of the two given circles as foci. This is, of course, equivalent to the definition that the sum or difference of the focal radii is constant. I have called such conics "associated" conics.

8 In what follows circles will be designated by their centers in brackets.

Pn

Xn

Let pn be the radius of (S) and let the coördinates of the point Sn be an and Yn. From a fundamental property of the ellipse we have

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Pappus has shown that if another circle (Sn+1) with radius pn+1, center at point 2n+1, yn+1 and coming after (Sn) in the positive direction around the circles, is tangent to (Sn), the following relation holds

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2. Formulæ arising from the figure of three mutually tangent circles with their centers in the same straight line. Let there be three mutually tangent circles (01, 02) and (03) having their centers in the same straight line and radii T1, T2, and rз respectively (Fig. 1). Then let a series of circles (S,) be drawn tangent to (01) and (O2), the first circle in the series being also tangent to (03) and each of the others tangent to the one preceding it in the series. There are two other such sets of tangent circles. The set tangent to (02) and (03) we will designate as (Sn), and the set tangent to (01) and (03) as (Sn"). Let the radii of the various sets be pn, pn' and pn" respectively, and the coördinates of the centers be xn, yn, Xn', yn' and xn", yn" respectively. We will now consider the set (Sn).

By means of equations (1), (2) and (3) and the use of induction we have in this particular case, since the y-coördinate of O3 = 0

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This result is arrived at by Muir and Fontené by different methods.2 Also from equations (2) and (4)

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is a convergent series, and if r3 approaches the limit 0, then (6) approaches the

value #r2/2 but is 0 at the limit.

1 Pappus, Collectio, p. 224.

2 See Introduction.

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