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This Elibron Classics book is a facsimile reprint of a 1830 edition by C.J.G. Rivington; etc., London. ... Read more

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A translation of Proclus's exposition of Euclid's methods and principles. This primary source for the history of mathematics contains much information about the work of mathematicians of the classical period. It is a rare work from antiquity which expounds the philosophy of mathematics. ... Read more

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Proclus, Euclid and the Platonic Theory of Mathematical Being
Proclus Diadochus (410-485 AD), whose vast literary output covers nearly the whole range of the Late Antique philosophical curriculum, here pays homage to the seminal work of Euclid in his 'Commentary on the Elements.' In addition to being a formal commentary and analysis on the rudiments of Euclid's 'Elements,' Proclus sets forth his philosophy of mathematics, which perhaps is the most important and compelling aspect of this work. Before referring to Proclus' theory of mathematics, it should be mentioned just what the commentator regarded as the meaning and purpose for Euclid's articulation of the 'Elements.' Proclus, in 'Elements' (2.68), states that "Euclid belonged to the persuasion of Plato...and this is why he thought the goal of the 'Elements' as a whole to be the construction of the so-called Platonic figures." These "Platonic figures" alluded to are the five regular solids, the basic cosmic elements, which are a major part of the discussion in the 'Timaeus' concerning the nature of the Demiurge's creation of the cosmos.So, ultimately the question Proclus answers about Euclid's objective for composing the 'Elements' is essentially that the author had in mind the Platonic Forms and the structure of the cosmos, not merely geometrical abstractions in isolation from three-dimensional bodies.We gather from this that Proclus' judgement not only views Euclid as a man of science who composed an introduction to geometry for academic instruction, but also elevates him to the status of a Platonic mystic illuminating cosmological and metaphysical mysteries.With that said, it is likely that Proclus' apotheosis of Euclid is somewhat extravagent and may well be unfounded, but it is very probable that Academic mathematical speculation and the 'Timaeus' text itself had some, if not a cardinal influence in Euclid's thought; and in that regard he may have been a Platonist.

Since Proclus sees Euclid as a mystic and a man of science, it is only necessary that his commentary on the 'Elements' reflect that image and be scientific and mystic itself.Just like Euclid's work, Proclus' is intended to be an instructional guide to--and to be an anylasis of--the science of geometry.But beyond the scope of academic formality, a higher goal pervades the entirety of Proclus' commentary; and thus it is requisite to briefly state his theoretical understanding on what the Platonic tradition designates as "mathematicals," or "mathematical being."The supreme principle, the One, is producer of all number via the principles of the Limit and the Unlimited. The Unlimited dynamic unfolds the inexhaustible power of the Infinite, exploding forth in a series of even numbers and is itself the cause the line. The co-ordinate dynamic of the Limit essentially neutralizes the Unlimited's propensity to infinity (which otherwise would proceed unchecked chaotically in an infinite series of even numbers), by offsetting, resovling and bounding the Even with the principle of Odd, which imposes finitely all Form and Order and Stability to the Indeterminate principle; and thus this principle is the cause of the circle. The result of the Limit and the Unlimited's activity is the production of Intelligible Being, known as the One-Being, the first member of the Noetic Triad [=Being, Life, Intellect]. Now, in the One-Being, it is axiomatic to the Platonic system that it possess all Being unitarily and all beings severally, meaning that everything there in the purely incorporeal One-Being is numbered determinately (thus limiting the concept of infinity to potency not actuality) and henceforth produces all number and being in Hypostatic Intellect, in Hypostatic Soul and in the en-Cosmic-Soul and the Cosmos itself.But when speaking of Ideal-Forms, they are not said to possess dimension, but are non-dimensional incorporeal forms; it is the process of projection and extension of the Forms into the Soul where all dimension transpires and is hence simaltaneously the locus for all mathematical being operating in cosmic phenomena.This means that in Proclus' account, "mathematicals" are prior to qualified three-dimensional bodies, as they have an intermediary status between their seat in Intellect and their operation in bodies.Moreover, mathematical being is, in short, co-existent with Soul, as inhering principally in its essence.This is why Proclus considers the whole science of geometry not to be mental abstractions from bodies, but rather as essential components of psychical forms which shape and qualify matter into three-dimensional objects that are only understood through the "eye" of the soul which discerns its own rational-principles and refers these back to the universal Ideal-Forms.

Ultimately, Proclus' theory of "mathematicals" is purely Platonic and is, in all probability, Pythagorean.And the idea of mathematical being, as something truly existent, is perhaps the most notoriously anti-Aristotelian doctrine in Platonism.Even though Aristotle did state that mathematics is about first causes, which are themselves divine [Metaphysics VI 1026.1a], he could not elevate numbers themselves to a state of the things that "truly exist" [see his rebuttal of Ideal-Numbers in Metaphysics bk.XIII, ch. 7].But it is up tointelligent readers and theorists to consider for themselves just what mathematics is and what its relation is to us and the universe.If readers are interested hearing a sublime philosophy of mathematics, the ancient voice of Proclus still has something say in this exegetical tribute to Euclid, which navigates both mystical and scientific number.

Proclus had deep geometric insight
I haven't read all of this book, only the parts relevant to Euclid's parallel postulate. Proclus had the idea for a proof of the latter, based on (A) Aristotle's Angle Unboundedness Axiom, and while that "proof" is flawed, it can be converted to a correct proof if a futher natural geometric hypothesis (S) is added. I will explain this in the forthcoming fourth edition of my text EUCLIDEAN and NON-EUCLIDEAN GEOMETRIES: DEVELOPMENT and HISTORY, where I name the result PROCLUS' THEOREM. It states that assuming Hilbert's axioms of incidence, betweenness and congruence, which fill most (but not all) of the gaps in Euclid, Euclid's Parallel Postulate is logically equivalent to the conjunction of (A) and (S).

Proclus was also many centuries ahead of his time in stating publicly that the angle of parallelism might not be a right angle - i.e., recognizing that non-Euclidean geometry might be possible, though he did not pursue that idea.

Good translation of Proclus
Proclus was the last major Neo-Platonic philosopher of importance before the Academy was closed by the Christian emperor Justinian in the 6th century.He wrote many works, including long commentaries on Plato's dialogues and a commentary on the Elements by Euclid.

This translation by Morrow, a leading Classicist, contains a good introductory essay on Proclus's philosophy of mathematics, along with other scholarly aids such as a biblography.While Proclus's ideas probably have little currency with either philosophers or mathematicians today, they are of interest to the student of Ancient philosophy and the historian of Philosophy. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Customer Reviews (3)

Love it
It's been 3 days and I haven't been able to put this book down. Absolutely fascinating. And of course the pictures are my favorite part.

Showplace of America: Cleveland's Euclid Avenue, 1850-1910 (Ohio)
This publication was exactly what I was looking for.Amazon found it --- and sent it to me quickly.

Cleveland History Fan's Delight
A wonderful trip back to Millionaire's Row, and the unbelievable homes that lined Cleveland's Euclid Avenue (including the fascinating men who built them), in the late 1800's.Plenty of pictures too.A must book forany lover of Cleveland history. ... Read more

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The immortal Elements of Euclid was already in antiquity a classic, regarded as absolutely perfect, valid without restriction. Elementary geolnmetry was for one thousand years as stationary, as fixed, as peculiarly Greek, as the Parthenon. On this foundation pure science rose in Archimedes, in Apollonius, in Pappus; struggled in Theon, in Hypatia; declined in Proems; fell into the long decadence of the Dark Ages. The book that monkish Europe could no longer understand was then taught in Arabic by Saracen and Moor in the Universities of Bagdad and Cordova. T0 bring the light, after weary, stupid centuries to Western Christendom, an Englishman, Adelhard of Bath, Journeys, to learn Arahic, through Asia NIinor, through Egypt. back to Spain. Disguised as a Mohammedan student, he got into Cordova about 1120, obtained a Moorish copy of Euclid's Elements and made a translation from the Arabic into Latin.

About the Publisher

Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology.

Forgotten Books' Classic Reprint Series utilizes the latest technology to regenerate facsimiles of historically important writings. Careful attention has been made to accurately preserve the original format of each page whilst digitally enhancing the difficult to read text. Read books online for free at www.forgottenbooks.org ... Read more

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Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This volume is produced from digital images created through the University of Michigan University Library's preservation reformatting program. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the text that can both be accessed online and used to create new print copies. This book and thousands of others can be found in the digital collections of the University of Michigan Library. The University Library also understands and values the utility of print, and makes reprints available through its Scholarly Publishing Office. ... Read more

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Originally published in 1920, this volume contains book one of Euclid's Elements in Greek, together with an introduction and notes section written in English. It was created to provide young students with a contextual understanding of geometry and the development of geometrical principles, one which was increasingly neglected in the standard school textbooks of the time. By returning to the Elements in their original form it was hoped that students would gain a fundamental understanding of the ideas put forward in the text, one which would increase their knowledge and enthusiasm. It was also hoped that the dual process of learning Greek and geometry would be an effective way of impressing content on the mind of the learner. This volume will be of value to anyone with an interest in geometry and the development of pedagogy. ... Read more

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This is an OCR edition without illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from the publisher's website (GeneralBooksClub.com). You can also preview excerpts of the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Subjects: History / General; ... Read more

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Euclid Golf Neighborhood reveals in vintage images the excellent planning and history of one of the finest neighborhoods in the country. Euclid Golf was built on land owned by John D. Rockefeller, who lent it to the Euclid Club for its golf course. Developer Barton R. Deming employed Garden City principles and deed restrictions to entice the elite of Cleveland, while architects Howell and Thomas, Charles Schneider, and others designed splendid houses in the revival styles that defined gracious living. As prominent Clevelanders made their homes in Euclid Golf, Fairmount Boulevard became known as "The Euclid Avenue of the Heights." ... Read more

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Euclid Golf
A very well done historical perspective of this development; was an offshoot of an application to get the area placed on the historic register. ... Read more

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With A Set Of Improved Figures, And An Original Demonstration Of The Proposition Called In Euclid The Twelfth Axiom. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Take a walk down Euclid Avenue from 1920 through 1970. The book takes a tour through the glory years, including a look at department stores such as Halle's, Taylor's, May's and Sterling Lindner Davis. Also included is a look at specialty shops like Bonwit Teller, Richman Brothers and Cowell and Hubbard Jewelers. You will read about restaurants such as the Halle Tea Room and Stouffer's, as well as the 11 theatres between East 17th and Public Square. The book also includes 22 recipes from Halle's Tea Room and Stouffer's restaurants.

Euclid Avenue, in the period of it glory as Cleveland's elegant main street, reflected the optimism, wealth, and opulence that characterized the city at the dawning of the twentieth century. Cleveland, Ohio, was then a city on the grow, a thriving metropolis, emerging as one of the nation's industrial and commercial powerhouses. ... Read more

Customer Reviews (2)

A must for Clevelanders
If you grew up in the Greater Cleveland Area, you'll be drawn back to the "golden age" of Euclid Avenue.It's all there... the department stores, the restaurants, the shops and stores, even the giant Christmas Tree at Sterling Linder Davis.It's amazing how much stuff I forgot, that came charging back when I read through this book.I highly recommend it.

Cleveland's Golden Age
A stroll down Euclid Avenue in the Jazz Age and beyond, at a time it was packed with top-tier restaurants, ornate theaters, and appealing stores, becomes pleasant in the hands of James Toman and Richard Karberg.Toman and Karberg have worked together before; they produced a wildly popular pair of books about Cleveland's venerable Silver Grille restaurant.

Toman--the author, coauthor, or editor of around 20 books of local interest--puts his usual strengths on display here.He chocks his books full of appealing photographs, combing through the archives, one suspects, to get just the right ones.The text is tightly written, carefully edited, and filled with anecdote, but it lets the pictures do much of the talking.

Cleveland is a decent respectable city of half a million now.Our art museum and orchestra are nationally prominent.Our hospitals, particularly UH and The Clinic, are among the country's best.But there was a time we were a million strong and on the grow, the future opening giddily before us.We were home to Rockefellers and a burgeoning industrial giant, and Euclid Avenue became a microcosm of that success spiced with a bit of "Great Gatsby"-style hedonism.To dip into this book is to date a "Sophisticated Lady," to feast the senses on the mink stoles, early film classics, and luscious cuisine of Cleveland's golden age. ... Read more