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THE CELEBRATED

THEORY OF PARALLELS.

DEMONSTRATION

OF

THE CELEBRATED THEOREM.

EUCLID I, AXIOM 12.

With Appendix containing the philosophy of the demonstration, together with the partial
refutation of Sir Wm. Hamilton's philosophy of the

Unconditioned or Infinite.

BY MATTHEW RYAN,

OF COUNTY TIPPERARY, IRELAND,

Clerk, War Department, (Office of Accounts, Gen. Chauncey McKeever,) Washington, D. C.,
Late 3d Reg't U. S. Infantry.]

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But you forget that Geometrical Equality can do great things, both among gods and men."

-Plato. Gorgias.

GRAND

HIATUS

FINITE

Thus, also, universally, from the comparison of the equality of finites may be evolved some positive knowl-
edge of the corresponding homogeneous Infinites, whether in Deity. space, time or degree.
-Appendix, Note A, Par. c.

WASHINGTON, D. C.:

CHRONICLE PRINT..

1866.

183. e. 11.

Entered according to Act of Congress, in the year 1865,

BY MATTHEW RYAN,

In the Clerk's Office of the District Court of the United States for the District of Columbia.

OTHEC

ROMANTIC

SKETCH

OF THE

HISTORY

ISTORY OF PARALLELS.

(a.) Of all contested geometrical subjects that of Parallels-the great blemish in the immortal "Elements" of Euclid-is the most ancient as well as the most fascinating. At the porch of science and clasping infinite SPACE, the subject is at once simple and sublime. Its recorded history commences with the mild and the benevolent Euclid, 280 years B. C. (see Euc. I, 28;) and though forming an essential element in most demonstrations, yet, as an unestablished theory, it must have been handed down by the illustrious Pythagoras, by the venerable Thales, and probably by the Egyptian priests who were the instructors of these philosophers.

* *

*

*

*

(b.) Says Lardner, "The Theory of Parallels has always been considered as the reproach of Geometry.* In pamphlet published 1856, T. P. Thompson † declares, “ Ptolomey, Proclus, the Arab editor of Euclid, Clavius, Wolfius, Boscovich, D'Alembert, Thomas Simpson, Bonnycastle, Robert Simson, Varignon, Bezout, Leslie, Ludlam, Playfair, Franceschini, Legendre, La Grange, Le Croix, Bertrand, with others of later date have successively set their seals to the admission that till something was done on this point‡ Geometry was not entitled to the name of an exact science."

(c.) Stevelly, (Lond. Edin. and Dub. Phil. Mag. 1856, Vol. XII, p 220,) frankly acknowledges thus: "I was again induced to waste some hours on a subject on which in my schoolboy and college days I, in common I suppose with every schoolboy and collegian since the days of Euclid, had over and over again wearied myself in vain."

*

(d.) During the past half century the matter had been subjected to the most subtle analysis; the doctrine of functions, and the inadmissible doctrine of limits had been appealed to, but in vain. In the investigation of the subject, indeed, "many have lost much time, and some even their reason. ** Its investigations have embittered the most exalted friendships. In this research the philosophic minds of Leslie and Legendre have waged war. Finally many moderns concluded that "to deduce the property of two lines postulated as parallels involves a direct dealing with the positive idea of infinity-a task utterly beyond the reach of our faculties," &c. || Similar, but more elaborate declarations will be found in an article on parallels by James Adamson, D. D., in the London, Edinburgh and Dublin Phil. Mag., 1853, Vol. V, page 407; also in the Penny Cyclopædia, Vol. XVII, art. parallels, concluding paragraph, page 238, as also in other works. Thus at length did this mysterious subject powerfully aid in the founding of a false and atheistical philosophy,¶¶ and bid fair to remain a matter of contention during all the ages of the future. But God has been pleased to hearken to the voice

*Dionysius Lardner, LL.D.-F. R. S-L. & E. (Geo. 1838.)

†General T. Perronet Thompson, of Eliot Vale, Blackheath, London.

Meaning parallels.

SEncyclopedia Americana.-Subject, Parallels.

Lond. Edin. and Dub. Phil. Mag. 1857, Vol. XIII. page 413, as quoted from "Nichols' Cyclopædia of the Physical Sciences."

Thus, on the hypothesis that a positive idea of the Infinite is impossible to the human mind, false philosophers have contended that hence the existence of an Infinite Being is unsusceptible of demonstration, and is therefore a superfluity, and as such, an evil as an element of man's religious belief.

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