irrational quantity as the area of a circle and equal to a parallelogram and convertible into a square by the usual rules, but not a square by his New Law in Geometry. That square will forever lack one square unit, however infinitesimal the measure-unit may be assumed.

In 1872, a work was published in London, entitled "A Budget of Paradoxes," by Prof. Augustus De Morgan, of Trinity College, Cambridge. It is composed of the collected articles, correspondence, reviews of books, etc., by Mr. De Morgan, published in the London Athenæum from 1863 to 1870. In this work of 512 pages there are mentioned the names of 75 writers on the subject of "Cyclometry." Mr. De Morgan has reviewed the works of 42 of these writers, giving the results of their search for the value of, bringing the subject down to 1870. The entire list has been compiled and tabulated by the writer of this monograph, which will accompany this paper. An examination of the compilation does not reveal the name of a single American author or book on the subject of " Cyclometry."

Of the 100 titles given in this bibliography, 52 are bound volumes, 32 are pamphlets, 7 are broadsides, and the remaining 9, including one manuscript, are communications to the press. These books have not been collected as a specialty, but are what naturally find their way on a variety of subjects into a mathematical collection of 700 or 800 volumes, and 500 or 600 pamphlets on "the bewitching science" of mathematics.

Those who desire to investigate these works, and the ingenious methods proposed to find the value of, can fully satisfy themselves that there are many roads to Rome. Many of the works are elaborate, and accompanied with artistic plates, and ample diagrams.

The results of the writers of the 100 titles are tabulated and classsified at the end, with other tables for comparison:

Errata. The following figures should be corrected in the following pages:

Page 110, sixth line from bottom, make the fifth decimal figure a 9. Page 112, eighth line from bottom, strike out the seventh decimal figure 3 from both decimals.

Page 130, tenth line from bottom, strike out the fifteenth and sixteenth decimal figures 46; also, insert figure 1, after the third figure 1 in the denominator of the fractional series.

Bibliography - Cyclometry and Quadratures.

ADORNO, JUAN NEPOMUCENO. Introduction to the Harmony of the Universe; or Principles of Physico-Harmonic Geometry. Plato said, "The Great Geometrician is God." The harmony of the universe proves the truth of this sublime sentence. Royal 8vo, cloth, pp. 160. 72 elaborate diagrams. London, 1851.

This is a very elaborate work on harmony, proportion, analogy, and ratio. The author says that he is convinced that "the circumference of any circle to its diameter is precisely as 22 to 7, a proportion considered by Archimedes as an approximation only." His ratio corresponds with that of William A. Myers, 3.1428574

ANGHERA, DOMENICO, REV. Quadratura del Cerchio. 8vo. cloth. Malta, 1858.

This priest says: "The circle is four times the square inscribed in its semicircle." Hence his area is .80, and his ratio is 3, or 3.2. BADDELEY, WILLIAM. Mechanical Quadrature of the Circle-- London Mechanics' Magazine, August, 1833.

"From a piece of carefully rolled sheet brass was cut out a circle 1 inches diameter, and a square 1 inches diameter. On weighing them they were found to be of exactly the same weight, which proves that, as each are of the same thickness, the surfaces must also be precisely similar. The rule, therefore, is that the square is to the circle as 17 to 19."

Hence this would make his ratio, 3.202216; area, .8005548T ANONYMOUS. Resumé for Analytic Exercise. 4to. Construction: To determine the point towards which an infinite descending series of triangles tend to a final term, being the limit of that segment of spiral which is the evolute of the quarter-circle.

To show the relations of lines representing the third root of quantities which are to each other as one and two, and the angle of the radius with the spiral which results from constructing the equation of the "two mean proportionals."

To transfer to any portion of the arc the conditions for its division into the same proportional parts as those of the semi-circle divided by the radius.

The author presents diagrams of circles and triangles combined and shows that the radius of one circle is " AB÷4," (which, if we understand him rightly,)=.26179387+, the ratio, 3.1415926535+.

BENNETT, JOHN. Original Geometrical Illustrations; or the Book of Lines, Square, Circles, Triangles, Polygons, &c., showing an easy and scientific analysis for increasing, decreasing, and altering any given circle, square, triangle, ellipse, parallelogram, polygon, &c., to any other figure containing the same area, by plain and simple methods, laid down agreeably to mathematical demonstrations; intended as a complete instructor to the most useful science of Geometry and Mensuration. 4to. cloth, pp. text, 70; plates, 54. Frontispiece, a diagram-The Circle, Square, and Triangle-primitive geometrical figures. London, 1837.

[Second Book.] The Arcanum, Comprising a Concise Theory of Practical Elementary and Definite Geometry; exhibiting the Various Transmutations of Superfices and Solids; obtaining also their Actual Capacity by the Mathematical Scale; including Solutions to the yet Unanswered Problems of the Ancients-The Circle, Square, and Rectangle of Similar Areas. 8vo. cloth. pp. 48. 176 diagrams. Frontispiece, a diagram-The Problem of Napoleon Buonaparte to his Staff, resolved and drawn by John Bennett. London, 1838.

Mr. Bennett says that the problem of corresponding areas of the square and circle "has remained altogether in obscurity; although rewards were offered by Charles V, of 1000 crowns; and the States of Holland a similar sum, to any person effecting it; but it does not appear to ever have been performed." He quadrates the circle thus: "The transverse of the circle being divided into 26 equal parts, 21 of those parts are found to occupy one-fourth of the circumference." Then he constructs the equi-areal square by intersecting the circumference at the 8 points of the 84 parts in the circumference, leaving 12 parts without the circumference and then 9 parts within the cir. cumference, alternately. This is a mechanical quadrature, and give for a ratio,, or 3.230769; area, .80769211, which is not in accord with his elaborate and artistic diagrams throughout his works. BENSON, LAWRENCE SLUTER. Scientific Disquisitions concerning the Circle and Ellipse; a Discussion of the Properties of the Straight Line and the Curve, with a critical examination of the Algebraic Analysis. "If a better system's thine, impart it frankly, or make use of mine." 12mo. cloth, pp. 94. Aiken, S. C., 1862.

Prof. Benson has published some twenty pamphlets, more or less on the area of the circle, three volumes of philosophic essays, and one geometry-"The Elements of Euclid and Legendre, Excluding the Reductio ad Absurdum. Reasoning." He endeavors to demonstrate

that the area of the circle is equal to 3R2, or the arithmetical square between the inscribed and circumscribed squares. His theorem is: "The 123.4641016+ is the ratio between the diameter of a circle and the perimeter of its equivalent square." The ratio between the diameter and circumference, he believes, is not a function of the area of the circle. He accepts the value of = 3.141592; but the area of the O, he believes, is 3R2, or .75.

BROWER, WILLIAM., M. D. The Quadrature of the Circle; being a full Exposition of the Problem. 8vo. pamphlet, pp. 16. 4 plates. Philadelphia, 1874.

The entire pamphlet is devoted to geometrical constructions and algebraic equations. According to his demonstration, he says:

"The circle is equal to the inscribed square,+2(side of inscribed square)X(width of quadrantal segment), +2(side of inscribed octagon) X(width of octagonal segment),-2(width of octagonal segment) X (side of inscribed square-side of inscribed octagon-width of quadrantal segment)."

Dr. Brower is ingenious and goes through many demonstrations, but the two triangles which he calls analogues are not similar, as he supposes, and therein lies his error. His first trial for the ratio results in 3.152075+, which he finds greater than the accepted ratio, and he concludes that a certain segment is less than x, an unknown quantity.

BOYAI, JANOS. . La science absolute de l'espace. 8vo. Paris, 1868.

We have never seen this work.

BOUCHE, CHARLES P. The Regulated Area of the Circle, and the Area of the Surface of the Sphere. 8vo. pamphlet, pp. 64. Cincinnati, Ohio, 1854.

Mr. Bouché says in the year 1823 he fixed the ratio to be 31, or 3.1604938271; but "in 1833 he found himself constrained to correct it to 3.1684; and later he found himself compelled to correct this." He finally made the ratio, 3.17124864+. The first ratio (31) was first developed by M. de Fauré in his "Dissertation, Découverte, et Démonstrations de la Quadrature Mathématique du Cercle," Geneva, 1747; and “Analyse de la Quadrature du Cercle," Hague, 1749, mentioned by De Morgan in his " Budget of Paradoxes," p. 89. same ratio was developed by Theodore Faber in 1865.

B., G. W. Squaring the Circle ; the Exact Circumference. Manufacturer and Builder, Vol. III, No., 2, p. 31, February, 1871, Mr. G's constructed diagram, and proposition amounts to this: "When we draw an equilateral triangle, of which each side is equal to the diameter of a given circle, one-fourth of the circumference will be equal to the radius plus one-third of the perpendicular of this triangle."

This proposition gives for the ratio, 3.1547005+, a number larger than a circumscribed polygon of 48 sides.

CARRICK, ALICK. The Secret of the Circle; its Area Ascertained. Second edition. 8vo. pp. 48. London, 1876.

Mr. Carrick, with the help of ten diagrams, some colored, concludes that the ratio is 34, or 3.1428574; and its area, .7857144. He ends his essay with these words; Patet omnibus veritas, multam ex illâ etiam futuris relicta est.

CART, FRANCIS GUERIN. The Problem of Centuries; What is the True Relation of Circumference to Diameter ? Diagrams.-News and Courier, (Charleston, S. C.) August 2, 1876.

Mr. Cart denies the universal correctness of the " 47th of Euclid." He took out a copyright of his new discoveries, September 14, 1875. under the title, "New light on an old subject, or an analysis of the present received science of geometry, showing its errors and revealing the truth." His proposition is as follows:

"The area of the circle is equal to the area of its circumscribed square minus the area of a rectangle whose hight is one-half the radius and whose side is the altitude of an equilateral triangle having the diameter for its base."

Hence his area of circle is 1-175, or .78349364+; this makes the ratio, 3.13397456+. He deduces it from 32, or 3.1339745+

CARTER, R. KELSO CAPT.

The Quadrature of the Circle; an Answer to Prof. Lawrence S. Benson's Proof that the Area of the Circle is Equal to Three Times the Square of the Radius. 8vo. pamphlet, pp. 16. Chester, Pa., 1876.

Capt. Carter says,

"Prof. Benson's method of proof is so ingeniously conducted that a very close study is necessary to discover its fallacy.' Prof. Benson believes that the area of a circle is equal to 3R2, or .75, and that the area is not a function of the ratio.