Book:Euclid/The Elements/Book VIII

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Euclid: The Elements: Book VIII

Published $\text {c. 300 B.C.E}$.


Contents

Book $\text{VIII}$: Theory of Proportions as applied to Number Theory

Proposition $1$: Geometric Sequence with Coprime Extremes is in Lowest Terms
Proposition $2$: Construction of Geometric Sequence in Lowest Terms
Porism to Proposition $2$: Construction of Geometric Sequence in Lowest Terms
Proposition $3$: Geometric Sequence in Lowest Terms has Coprime Extremes
Proposition $4$: Construction of Sequence of Numbers with Given Ratios
Proposition $5$: Ratio of Products of Sides of Plane Numbers
Proposition $6$: First Element of Geometric Sequence not dividing Second
Proposition $7$: First Element of Geometric Sequence that divides Last also divides Second
Proposition $8$: Geometric Sequences in Proportion have Same Number of Elements
Proposition $9$: Elements of Geometric Sequence between Coprime Numbers
Proposition $10$: Product of Geometric Sequences from One
Proposition $11$: Between two Squares exists one Mean Proportional
Proposition $12$: Between two Cubes exist two Mean Proportionals
Proposition $13$: Powers of Elements of Geometric Sequence are in Geometric Sequence
Proposition $14$: Number divides Number iff Square divides Square
Proposition $15$: Number divides Number iff Cube divides Cube
Proposition $16$: Number does not divide Number iff Square does not divide Square
Proposition $17$: Number does not divide Number iff Cube does not divide Cube
Proposition $18$: Between two Similar Plane Numbers exists one Mean Proportional
Proposition $19$: Between two Similar Solid Numbers exist two Mean Proportionals
Proposition $20$: Numbers between which exists one Mean Proportional are Similar Plane
Proposition $21$: Numbers between which exist two Mean Proportionals are Similar Solid
Proposition $22$: If First of Three Numbers in Geometric Sequence is Square then Third is Square
Proposition $23$: If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube
Proposition $24$: If Ratio of Square to Number is as between Two Squares then Number is Square
Proposition $25$: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
Proposition $26$: Similar Plane Numbers have Same Ratio as between Two Squares
Proposition $27$: Similar Solid Numbers have Same Ratio as between Two Cubes


Critical View

Once the more-or-less profound results about the structure of a geometric sequence of integers has been comprehended, the rest of the book consists of simple examples.

Hence, once the initial few results have been established in contemporary language (and to as high a power as is desired), the Euclidean proofs of these results, and indeed the results themselves, become all but irrelevant.