ROLLWYN, J. A. S. Elementary Difficulties in Geometry: The Duplication of the Cube; The Trisection of an Angle; The Quadrature of the Circle. Chapter XXXIII, of " Astronomy Simplified, for general reading with numerous New Explanations and Discoveries." London, 1871. 8vo. pp. 10.

"The area of a circle is equal to three-fourths of the square of its diameter, or three-fourths of the area of the circumscribed square; and that concurrently, twice the area of the circle is equal to three times the area of the inscribed sqnare."

Rule-Multiply the diameter of the circle by itself and deduct onefourth of the product; the remaining quantity is the area of the circle. Mr. Rollwyn's area is the same as that of Prof. L. S. Benson-that it is the arithmetical square between the inscribed and circumscribed squares.

Rossi, GAETANO, of Catanzaro.

Soluzione Esatta, e Regolare deDifficillissimo Problema della Quadratura del Circolo; Produzzione Sintetica, ed Analitica. Hæc qui spernit, id esi Semitas Sapentiæ, ei denuncio non recte philosophandum.—BOETIUS. Seconda edizione. Londra, 1805. 8vo. pamphlet, pp. 108. 8 diagrams; portrait. The author's demonstrations result in the ratio, 3, and area, 80. SCHOLFIELD, NATHAN. On the Rectification and Quadrature of the Circle. Part Fourth of a Series on Elementary and Higher Geometry, Trigonometry, and Mensuration; containing many valuable Discoveries and Improvements in Mathematical Science, especially in relation to the Quadrature of the Circle, and some other Curves. 8vo. pp. 108-139. New York, 1845.

Mr. Scholfield's treatise is a learned and searching analysis on the subject of curves, segments, spirals, cycloids, revoloids, etc. He substantiates the orthodox ratio, 3.141592653589793238462643+.

SKINNER, J. RALSTON. A Criticism on the Legendre Mode of the Rectification of the Curve of the Circle. 8vo. pamphlet, pp. 22. Cincinnati, 1881.

The author says the orthodox value of pi obtained by the Legendre method from the sides of the interior polygons is numerical, and nor geometrical. The circumference of a circle is a curve which finally reenters on itself and forms the boundary of the circle. The numerical values of the polygons are not indicative of the circle penned up between them. Mr. Skinner's demonstrations substantiate the ratio as found by John A. Parker, namely, 2012, or 3.1415942+.

SHANKS, WILLIAM. Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Decimals. Royal 8vo. pp. 95. London, 1853.

Mr. Shanks here publishes the value of to 607 decimal plac es He gives the value of (Naperian base) to 137 decimal places, the value of M (Modulus) to 137 decimal places, and the powers of 2 as far as 2721. He was assisted by Dr. William Rutherford in the verification of the first 441 decimals of . Since the publication of this work, Mr. Shanks has found errors in the last 14 places of the 607 decimals, as printed in this book, corrected the errors, and then ex. tended the decimals to 707 places, and they are printed by the Royal Society of London, in their Proceedings, Vol. XXI, 1873, as follows:

3.141592 653589 793238 462643 383279 502884 197169 399375 105820 974944 592307 816406 286208 998628 034825 342117 067982 148086 513282 306647 093844 609550 582231 725359 408128 481117 450284 102701 938521 105559 644622 948954 930381 964428 810975 665933 446128 475648 233786 783165 271201 909145 648566 923460 348610 454326 648213 393607 260249 141273 724587 006606 315588 174881 520920 962829 254091 715364 367892 590360 011330 530548 820466 521384 146951 941511 609433 057270 365759 591953 092186 117381 932611 793105 118548 074462 379834 749567 351885 752724 891227 938183 011949 129833 673362 441936 643086 021395 016092 448077 230943 628553 096620 275569 397986 950222 474996 206074 970304 123668 861995 110089 202383 770213 141694 119029 885825 446816 397999 046597 000817 002963 123773 813420 841307 914511 839805 70985±

SMITH, JAMES. Relations of a Circle inscribed in a Square. pp. 6 Commensurable Relations between a Circle and other Geometrical Figures. 1860.

Quadrature of the Circle; Correspondence with an" Emi

Nut to Crack for the Readers of De Morgan's "Budget of
Paradoxes." 1863.

True Ratio between Diameter and Circumference, Geo-
metrically and Mathematically Demonstrated. 1865.
British Association in Jeopardy and Prof. De Morgan in
the Pillory without hope of escape. 1866.
Quadrature and Rectification of the Circle. 1867.
Euclid at Fault, in Theorem, Proposition 8, Book VI;

pp. 32

pp. 188

and Theorems, Propositions 12 and 13, Book II. 1868. pp.

pp. 416 pp. 98

SMITH, JAMES. Geometry of the Circle and Mathematics as
applied by Geometers and Mathematicians, shown to be a
Mockery, Delusion, and a Snare. 1869.
Curiosities of Mathematics, Instruction of Mathematicians.
The Ratio between Diameter and Circumference Demon-
strated by Angles, and Euclid's Theorem, Proposition
23, Book I, Proved to be Fallacious. 1870.
Cyclometry and Circle-Squaring in a Nutshell. 1871.
Why is Euclid Unsuitable as a Text-book of Geometry?
Theorems, Propositions of Euclid, 8 and 13, Book VI,
Proved to be Erroneous, by Heterodox Geometers. 1871. pp. 56
Quadrature and Geometry of the Circle Demonstrated.
Portrait. London and Liverpool, 1872.

pp. 524

pp. 44

pp. 268 These works are profusely illustrated with plates, diagrams, extracts, and examples. He demonstrates the ratio to be 25, or 3.125. He credits Joseph Lacomme with finding this ratio, in 1836, who is found in De Morgan's list. Mr. Smith's works totalize 1988 pages on this

subject.

SMITH, SEBA. New Elements of Geometry. Three Parts. I. The Philosophy of Geometry. II. The Demonstrations in Geometry. III. The Harmonies of Geometry. 8vo. pp. 200. New York, 1850. London edition, pp. 200, 1850.

Mr. Smith examined John A. Parker's manuscript quadrature, became convinced of the truth of it, and published his own Geometry the year previous to the publication of Mr. Parker's " Quadrature of the Circle."

SMOOTH, EPHRAIM. Measuring of Circles; the Proportion which the Diameter bears to the Circumference-Register of Arts and Sciences, July 8, 1826. London.

Mr. Smooth illustrates both his and Archimedes' ratio by examples, and claims that the ratio is 31.

SOMERSET, (Duke of). A Treatise in which the Elementary Properties of the Ellipse are deduced from the Propertise of the Circle, and geometrically considered. Illustrated. 8vo. London. 1843.

STACY, JOSEPH. Squaring the Circle.-Boston Herald, April 4, 1874. Mr. Stacy says he "has no more difficulty in obtaining the ratio than in obtaining the diagonal of a square. The circumference of a circle, as near as can be expressed in so many figures, is 3.152955+; the error is less than 1 in 75,000,000; or it makes a difference of 91 miles in the circumference, or 29 miles in the diameter of the earth."

STEELE, JAMES. Exact Numerical Quadrature of the Circle effected regardless of the Circumference, and the Commensurability of the Diagonal and Side of the Square. 8vo. pp. 75. London, 1881. The author finds the exact area of a circle in the nonary scale, but "The circle is equal to 9 square units When the circumscribed square is = We do not comprehend this statement.

just how he does not explain:
nonarily expressed as 10.
the circle is 1.570796+."

= 2,

TAGEN, JOANNEM Nep. Quadratura circuli tandem inventa, et mathematice demonstrata; cum II tabulis. Folded diagrams. 8vo. pp. 75. Cassoviæ, 1832. [His methods are very complex.]

TERRY, CONSTANT. A Problem for the World; the Circle Squared. Eagle Pass, Texas, January 20, 1871. Published in The Investigator, Boston, Mass., February 22, 1871.

"I demand the solution of a circle whose area, diameter, and circumference are each perfect squares."

I.

2.

3.

4.

5.

Circumference 1, diameter=.316912650057057350374175801344.
Circumference 1, area .79228162514264337593543950336.
Diameter 1, circumference 3.1554436208840472216469142611-

3114491869282574043609201908111572265625.

Area 100, diameter 11.25899906842624.

Area 100, circum.=35.52713678800500929355621337890625. 6. Area 100

7.

circumference

diameter.

Multiply area, when circumference is 1, by area, when diameter is 1, and the product is .0625.

8. Multiply diameter, when circumference is 1, by the circumference, when diameter is 1, and the product is 1.

9.

Multiply square of diameter by square of circumference, when

area is 100, and the product is 160,000.

Square root of (1) is .562949953421312.

Square root of (2) is .281474976710656.

Square root of (3) is 1.7763568394002504646778106689453125. Square root of (4) is 3.3554432.

Square root of (5) is 5.9604644785390625.

THOMPSON, G. H.

The Discovery of the Quadrature; announced to the world by the Divine Assistance.-In Coram's Champion, 1826. Mr. Thompson's quadrature was the forerunner of that developed by Augustus Young nineteen years later in his first edition of "Ration

al Analysis," 1845. The prime formula upon which it was based is

We do not comprehend this formula as published in Scientific Tracts and Family Lyceum, Vol. I, p. 157, by Augustus Young, the champion of Mr. Thompson.

THORNTON, EDWARD. The Circle Squared. 8vo. London, 1868. Mr. Thornton's quadrature agrees precisely with Lawrence S. Benson's, in making the circle-area, 3 R2, or .75.

UPTON, WILLIAM, B. A. The Circle Squared. Three famous Problems of Antiquity, Geometrically Solved. The Quadrature of the Circle; Diameter definitely expressed in terms of the Circumference; Circumference equalized by a Right Line. The whole rendered intelligible for arithmeticians as well as for geometers; adapted for the higher classes in schools of both sexes, private students, collegians, &c. "Mutans quadrata rotundis."-HORACE. 8vo. pamphlet, pp. 24. Supplement: The Circle Squared; First Arithmetical Summary; Second-Geometrical Confirmation. opus." Plates. pp. 8. London, 1872.

"Finis coronat

The author demonstrates the orthodox ratio, 3.14159265+, by several methods not found in our text-books.

VANDERWEYDE, PHILIP H., M. D. The Philosopher's Stone: Four Essays, containing the Answer of Positive Science to the Question, What is known at present, about the Quadrature of the Circle? 8vo. pamphlet, pp. 40. New York, 1861.

Dr. Vanderweyde, for many years editor of The Manufacturer and Builder, has given in this essay an epitomized account of what the subject is, and then endeavors to answer it. The rectification of the circle is answered by several methods of demonstration, resulting in the ratio 3.1415926535+.

WEATHERBY, J. G. To find a Straight Line equal to the Semi-circumference of the Circle.-Barnes' Teacher's Monthly, Vol. I, p. 384, July, 1875.

Mr. Weatherby's geometrical construction and equation makes the semi-circumference of a circle of diameter 60, to be 94.6 (nearly). Hence, the ratio, 3.15333 (nearly).