Elements, first six books (Casey)
Euclid, c. 300 BCE
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The Elements builds geometry from a short list of definitions, postulates and common notions, then proves each proposition from those and from earlier propositions. This edition, by John Casey, carries the first six books and the first propositions of Book XI. Book I treats angles, triangles, parallels and parallelograms, and ends with the Pythagorean theorem. Book II handles rectangles, Book III the circle, Book IV figures inscribed in and drawn around circles, Book V proportion, Book VI similar figures, and Book XI planes and solid angles. Casey adds notes and exercises for students.
Where it happens
- Alexandria Euclid is placed here around 300 BC, where the Elements was written.2
- Oxyrhynchus Findspot of a papyrus fragment of the Elements with a diagram for Book II, Proposition 5.1
- Venice Ratdolt printed the first Latin edition here in 1482.1
- Basel Grynaeus printed the first Greek edition here in 1533.1
- Oxford The Bodleian Library holds copies of the Greek text.1
- Dublin Casey signed his preface to this edition here in 1885.3
Through time
- 300 BCEuclid writes the Elements in Alexandria, in thirteen books.12
- 100A papyrus fragment of the Elements from Oxyrhynchus is dated to about this time.1
- 800Arabic translations of the Elements appear in the ninth century.1
- 1482Ratdolt prints the first Latin edition in Venice.1
- 1533Grynaeus prints the first Greek edition in Basel.1
- 1570Billingsley's first English edition appears with a preface by John Dee.1
- 1885Casey signs the preface to his edition of the first six books.3
Who is in it, and behind it
John Casey translator
Irish editor of this edition, who signed his preface in Dublin in November 1885.3
Proclus subject
Greek mathematician whose commentary on the Elements is a main source on its background.1
Theon of Alexandria scribe
Edited the Elements in the fourth century, and later Greek copies descend from his version.1
Arethas patron
Byzantine scholar who commissioned a Greek manuscript of Euclid in the late ninth century, one of the two oldest surviving.1
Adelard of Bath translator
Made a twelfth-century Latin translation of the Elements from Arabic.1
Erhard Ratdolt printer
Printed the first Latin edition of the Elements in Venice in 1482.1
John Dee subject
Wrote the Mathematical Preface to Henry Billingsley's English edition of 1570.1
What it is about
Proof from stated starting points
Each result rests on listed definitions, postulates and axioms, so a reader can check every step back to them.13
The parallel postulate
The fifth postulate is less obvious than the first four, and questions about it led to non-Euclidean geometry.1
Geometry without numbers
Magnitudes of shapes are compared through constructions rather than treated as numbers.1
Theory of proportion
Book V treats ratio for any kind of magnitude, and Book VI applies it to similar figures.3
In its own words
Geometry is the Science of figured Space. Figured Space is of one, two, or three dimensions, according as it consists of lines, surfaces, or solids
A point is that which has position but not dimensions. A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line
Axioms. i. Things which are equal to the same, or to equals, are equal to each other
PROP. V.—Theorem. The angles (ABC, ACB) at the base (BC) of an isosceles triangle are equal to one another, and if the equal sides (AB, AC) be produced, the external angles (DBC, ECB) below the base shall be equal
PROP. XLVII.—Theorem. In a right-angled triangle (ABC) the square on the hypotenuse (AB) is equal to the sum of the squares on the other two sides (AC, BC)
Ratio is the mutual relation of two magnitudes of the same kind with respect to quantity
Euclid’s definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections); and even for these the modern definition is better
Similar Rectilineal Figures are those whose several angles are equal, each to each, and whose sides about the equal angles are proportional
The shape of the book
11 chapters, 87,446 words. Point at a bar.
How it came to be here
Euclid compiled the Elements around 300 BC, and it is the oldest large deductive treatment of mathematics that survives. Proclus wrote a commentary on it, and Theon of Alexandria edited it in the fourth century. Arabic translations followed in the ninth century, and Adelard of Bath made a Latin version from Arabic in the twelfth. Ratdolt printed the first Latin edition in Venice in 1482, and Grynaeus the first Greek one in Basel in 1533. Casey dated his preface from Dublin in November 1885, writing for Irish colleges and schools.
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Sources
- Euclid's Elements - Wikipedia
- Euclid - Wikipedia
- The First Six Books of the Elements of Euclid by Euclid and John Casey - Project Gutenberg
Text: PG 21076 | translator: John Casey. US public domain (Project Gutenberg licence).