veritas – 페이지 3

2018년 12월 20일2018년 12월 20일

7권

Definitions

Definition 1: A unit is that by virtue of which each of the things that exist is called one.
Definition 2: A number is a multitude composed of units.
Definition 3: A number is a part of a number, the less of the greater, when it measures the greater;
Definition 4: But parts when it does not measure it.
Definition 5: The greater number is a multiple of the less when it is measured by the less.
Definition 6: An even number is that which is divisible into two equal parts.
Definition 7: An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.
Definition 8: An even-times-even number is that which is measured by an even number according to an even number.
Definition 9: An even-times-odd number is that which is measured by an even number according to an odd number.
Definition 10: An odd-times-odd number is that which is measured by an odd number according to an odd number.
Definition 11: A prime number is that which is measured by a unit alone.
Definition 12: Numbers relatively prime are those which are measured by a unit alone as a common measure.
Definition 13: A composite number is that which is measured by some number.
Definition 14: Numbers relatively composite are those which are measured by some number as a common measure.
Definition 15: A number is said to multiply a number when the latter is added as many times as there are units in the former.
Definition 16: And, when two numbers having multiplied one another make some number, the number so produced be called plane, and its sides are the numbers which have multiplied one another.
Definition 17: And, when three numbers having multiplied one another make some number, the number so produced be called solid, and its sides are the numbers which have multiplied one another.
Definition 18: A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
Definition 19: And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
Definition 20: Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
Definition 21: Similar plane and solid numbers are those which have their sides proportional.
Definition 22: A perfect number is that which is equal to the sum its own parts.

Propositions

Proposition 1: When two unequal numbers are set out, and the less is continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, then the original numbers are relatively prime.
Proposition 2: To find the greatest common measure of two given numbers not relatively prime. Corollary. If a number measures two numbers, then it also measures their greatest common measure.
Proposition 3: To find the greatest common measure of three given numbers not relatively prime.
Proposition 4: Any number is either a part or parts of any number, the less of the greater.
Proposition 5: If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.
Proposition 6: If a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one.
Proposition 7: If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole.
Proposition 8: If a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole.
Proposition 9: If a number is a part of a number, and another is the same part of another, then alternately, whatever part or parts the first is of the third, the same part, or the same parts, the second is of the fourth.
Proposition 10: If a number is a parts of a number, and another is the same parts of another, then alternately, whatever part of parts the first is of the third, the same part, or the same parts, the second is of the fourth.
Proposition 11: If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole.
Proposition 12: If any number of numbers are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.
Proposition 13: If four numbers are proportional, then they are also proportional alternately.
Proposition 14: If there are any number of numbers, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
Proposition 15: If a unit number measures any number, and another number measures any other number the same number of times, then alternately, the unit measures the third number the same number of times that the second measures the fourth.
Proposition 16: If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another.
Proposition 17: If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied.
Proposition 18: If two number multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers.
Proposition 19: If four numbers are proportional, then the number produced from the first and fourth equals the number produced from the second and third; and, if the number produced from the first and fourth equals that produced from the second and third, then the four numbers are proportional.
Proposition 20: The least numbers of those which have the same ratio with them measure those which have the same ratio with them the same number of times; the greater the greater; and the less the less.
Proposition 21: Numbers relatively prime are the least of those which have the same ratio with them.
Proposition 22: The least numbers of those which have the same ratio with them are relatively prime.
Proposition 23: If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.
Proposition 24: If two numbers are relatively prime to any number, then their product is also relatively prime to the same.
Proposition 25: If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.
Proposition 26: If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime.
Proposition 27: If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime.
Proposition 28: If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime.
Proposition 29: Any prime number is relatively prime to any number which it does not measure.
Proposition 30: If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers.
Proposition 31: Any composite number is measured by some prime number.
Proposition 32: Any number is either prime or is measured by some prime number.
Proposition 33: Given as many numbers as we please, to find the least of those which have the same ratio with them.
Proposition 34: To find the least number which two given numbers measure.
Proposition 35: If two numbers measure any number, then the least number measured by them also measures the same.
Proposition 36: To find the least number which three given numbers measure.
Proposition 37: If a number is measured by any number, then the number which is measured has a part called by the same name as the measuring number.
Proposition 38: If a number has any part whatever, then it is measured by a number called by the same name as the part.
Proposition 39: To find the number which is the least that has given parts.

Guide

Book VII is the first of the three books on number theory. It begins with the 22 definitions used throughout these books. The important definitions are those for unit and number, part and multiple, even and odd, prime and relatively prime, proportion, and perfect number. The topics in Book VII are antenaresis and the greatest common divisor, proportions of numbers, relatively prime numbers and prime numbers, and the least common multiple.

The basic construction for Book VII is antenaresis, also called the Euclidean algorithm, a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger until a single number is left. This algorithm, studied in propositions VII.1 through VII.3, results in the greatest common divisor of two or more numbers.

Propositions V.5 through V.10 develop properties of fractions, that is, they study how many parts one number is of another in preparation for ratios and proportions.

The next group of propositions VII.11 through VII.19 develop the theory of proportions for numbers.

VII.11

a:b=c:d implies both equal the ratio a–c:b–d.

VII.12

a:b=c:d implies both equal the ratio a+c:b+d.

VII.13 .

a:b=c:d implies a:c=b:d.

VII.14

a:b=d:e and b:c=e:f imply a:c=d:f.

VII.17

a:b=ca:cb.

VII.19

a:b=c:d if an only if ad=bc.

Propositions VII.20 through VII.29 discuss representing ratios in lowest terms as relatively prime numbers and properties of relatively prime numbers. Properties of prime numbers are presented in propositions VII.30 through VII.32. Book VII finishes with least common multiples in propositions VII.33 through VII.39.

2018년 12월 20일2018년 12월 20일

6권

definitions (4)
propositions (33)

Definitions

Definition 1.: Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional.
Definition 2.: Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional.
Definition 3.: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
Definition 4.: The height of any figure is the perpendicular drawn from the vertex to the base.

Propositions

Proposition 1.: Triangles and parallelograms which are under the same height are to one another as their bases.
Proposition 2.: If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally; and, if the sides of the triangle are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle.
Proposition 3.: If an angle of a triangle is bisected by a straight line cutting the base, then the segments of the base have the same ratio as the remaining sides of the triangle; and, if segments of the base have the same ratio as the remaining sides of the triangle, then the straight line joining the vertex to the point of section bisects the angle of the triangle.
Proposition 4.: In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.
Proposition 5.: If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides.
Proposition 6.: If two triangles have one angle equal to one angle and the sides about the equal angles proportional, then the triangles are equiangular and have those angles equal opposite the corresponding sides.
Proposition 7.: If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles either both less or both not less than a right angle, then the triangles are equiangular and have those angles equal the sides about which are proportional.
Proposition 8.: If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the triangles adjoining the perpendicular are similar both to the whole and to one another.Corollary. If in a right-angled triangle a perpendicular is drawn from the right angle to the base, then the straight line so drawn is a mean proportional between the segments of the base.
Proposition 9.: To cut off a prescribed part from a given straight line.
Proposition 10.: To cut a given uncut straight line similarly to a given cut straight line.
Proposition 11.: To find a third proportional to two given straight lines.
Proposition 12.: To find a fourth proportional to three given straight lines.
Proposition 13.: To find a mean proportional to two given straight lines.
Proposition 14.: In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.
Proposition 15.: In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.
Proposition 16.: If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional.
Proposition 17.: If three straight lines are proportional, then the rectangle contained by the extremes equals the square on the mean; and, if the rectangle contained by the extremes equals the square on the mean, then the three straight lines are proportional.
Proposition 18.: To describe a rectilinear figure similar and similarly situated to a given rectilinear figure on a given straight line.
Proposition 19.: Similar triangles are to one another in the duplicate ratio of the corresponding sides.Corollary. If three straight lines are proportional, then the first is to the third as the figure described on the first is to that which is similar and similarly described on the second.
Proposition 20.: Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.Corollary. Similar rectilinear figures are to one another in the duplicate ratio of the corresponding sides.
Proposition 21.: Figures which are similar to the same rectilinear figure are also similar to one another.
Proposition 22.: If four straight lines are proportional, then the rectilinear figures similar and similarly described upon them are also proportional; and, if the rectilinear figures similar and similarly described upon them are proportional, then the straight lines are themselves also proportional.
Proposition 23.: Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
Proposition 24.: In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
Proposition 25.: To construct a figure similar to one given rectilinear figure and equal to another.
Proposition 26.: If from a parallelogram there is taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, then it is about the same diameter with the whole.
Proposition 27.: Of all the parallelograms applied to the same straight line falling short by parallelogrammic figures similar and similarly situated to that described on the half of the straight line, that parallelogram is greatest which is applied to the half of the straight line and is similar to the difference.
Proposition 28.: To apply a parallelogram equal to a given rectilinear figure to a given straight line but falling short by a parallelogram similar to a given one; thus the given rectilinear figure must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram.
Proposition 29.: To apply a parallelogram equal to a given rectilinear figure to a given straight line but exceeding it by a parallelogram similar to a given one.
Proposition 30.: To cut a given finite straight line in extreme and mean ratio.
Proposition 31.: In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
Proposition 32.: If two triangles having two sides proportional to two sides are placed together at one angle so that their corresponding sides are also parallel, then the remaining sides of the triangles are in a straight line.
Proposition 33.: Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences.

Logical structure of Book VI

Proposition VI.1 is the basis for the entire of Book VI except the last proposition VI.33. Only these two propositions directly use the definition of proportion in Book V. Proposition VI.1 constructs a proportion between lines and figures while VI.33 constructs a proportion between angles and circumferences. The intervening propositions use other properties of proportions developed in Book V, but they do not construct new proportions using the definition of proportion.

2018년 12월 20일2018년 12월 20일

5권

Definition 1

A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

크기는 크기의 부분(그것으로 더 큰 것을 측정했을 때, 큰 것의 작은) 부분이다. (magnitude는 크기로 옮기자.)

Definition 2

The greater is a multiple of the less when it is measured by the less.더 큰 것은 작은 것으로 측정된다면 작은 것의 배수이다.

Definition 3

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.비율은 같은 종류의 크기 사이에서 치수에 관한 관계의 한 종류이다. (size는 크기를 잰 값으로 치수로 옮긴다.)

Definition 4

Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.크기는 어떤 것의 적당한 배수로 다른 것을 넘어 설 수 있다면 비율을 가진다고 말한다.

Definition 5

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.다음을 만족하면 크기는 1에서 2까지 그리고 3에서 4까지는 같은 비율이라고 부른다. 1과 3으로 결정되는 같은 배수와 2와 4로 정해지는 같은 배수가 있을 때, 전자가 후자를 똑같이 넘어 서거나, 둘이 같거나, 똑같이 잛다.

Definition 6

Let magnitudes which have the same ratio be called proportional.

Definition 7

When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

Definition 8

A proportion in three terms is the least possible.

Definition 9

When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

Definition 10

When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

Definition 11

Antecedents are said to correspond to antecedents, and consequents to consequents.

Definition 12

Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.

Definition 13

Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.

Definition 14

A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself.

Definition 15

A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.

Definition 16

Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.

Definition 17

A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, the first is to the last among the first magnitudes as the first is to the last among the second magnitudes. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.

Definition 18

A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, antecedent is to consequent among the first magnitudes as antecedent is to consequent among the second magnitudes, while, the consequent is to a third among the first magnitudes as a third is to the antecedent among the second magnitudes.

Propositions

Proposition 1

If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.

Proposition 2

If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.

Proposition 3

If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.

Proposition 4

If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

Proposition 5

If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.

Proposition 6

If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them.

Proposition 7

Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio.Corollary If any magnitudes are proportional, then they are also proportional inversely.

Proposition 8

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

Proposition 9

Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal.

Proposition 10

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

Proposition 11

Ratios which are the same with the same ratio are also the same with one another.

Proposition 12

If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.

Proposition 13

If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth.

Proposition 14

If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.

Proposition 15

Parts have the same ratio as their equimultiples.

Proposition 16

If four magnitudes are proportional, then they are also proportional alternately.

Proposition 17

If magnitudes are proportional taken jointly, then they are also proportional taken separately.

Proposition 18

If magnitudes are proportional taken separately, then they are also proportional taken jointly.

Proposition 19

If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole.Corollary. If magnitudes are proportional taken jointly, then they are also proportional in conversion.

Proposition 20

If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less.

Proposition 21

If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less.

Proposition 22

If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.

Proposition 23

If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali.

Proposition 24

If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth.

Proposition 25

If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.

Guide for Book V

Background on ratio and proportion

Book V covers the abstract theory of ratio and proportion. A ratio is an indication of the relative size of two magnitudes. The propositions in the following book, Book VI, are all geometric and depend on ratios, so the theory of ratios needs to be developed first. To get a better understanding of what ratios are in geometry, consider the first proposition VI.1. It states that triangles of the same height are proportional to their bases, that is to say, one triangle is to another as one base is to the other. (A proportion is simply an equality of two ratios.) A simple example is when one base is twice the other, therefore the triangle on that base is also twice the triangle on the other base. This ratio of 2:1 is fairly easy to comprehend. Indeed, any ratio equal to a ratio of two numbers is easy to comprehend. Given a proportion that says a ratio of lines equals a ratio of numbers, for instance, A : B = 8:5, we have two interpretations. One is that there is a shorter line CA = 8C while B = 5C. This interpretation is the definition of proportion that appears in Book VII. A second interpretation is that 5 A = 8 B. Either interpretation will do if one of the ratios is a ratio of numbers, and if A : B equals a ratio of numbers that A and B are commensurable, that is, both are measured by a common measure.

Many straight lines, however, are not commensurable. If A is the side of a square and B its diagonal, then A and B are not commensurable; the ratio A : B is not the ratio of numbers. This fact seems to have been discovered by the Pythagoreans, perhaps Hippasus of Metapontum, some time before 400 B.C.E., a hundred years before Euclid’s Elements.

The difficulty is one of foundations: what is an adequate definition of proportion that includes the incommensurable case? The solution is that in V.Def.5. That definition, and the whole theory of ratio and proportion in Book V, are attributed to Eudoxus of Cnidus (died. ca. 355 B.C.E.)

Summary of the propositions

The first group of propositions, 1, 2, 3, 5, and 6 only mention multitudes of magnitudes, not ratios. They each either state, or depend strongly on, a distributivity or an associativity. In the following identities, m and n refer to numbers (that is, multitudes) while letters near the end of the alphabet refer to magnitudes.

V.1. Multiplication by numbers distributes over addition of magnitudes.

m(x₁ + x₂ + … + x_n) = m x₁ + m x₂ + … + m x_n.V.2. Multiplication by magnitudes distributes over addition of numbers.

(m + n)x = mx + nx.V.3. An associativity of multiplication.

m(nx) = (mn)x.V.5. Multiplication by numbers distributes over subtraction of magnitudes.

m(x – y) = mx – my.V.6. Uses multiplication by magnitudes distributes over subtraction of numbers.

(m – n)x = mx – nx.The rest of the propositions develop the theory of ratios and proportions starting with basic properties and progressively becoming more advanced.

V.4. If w : x = y : z, then for any numbers m and n, mw : mx = ny : nz.

V.7. Substitution of equals in ratios. If x = y, then x : z = y : z and z : x = z : y.

V.7.Cor. Inverse proportions. If w : x = y : z, then x : w = z : y.

V.8. If x < y, then x : z < y : z but z : x > z : y.

V.9. (A converse to V.7.) If x : z = y : z, then x = y. Also, if z : x = z : y, then x = y.

V.10. (A converse to V.8.) If x : z < y : z, then x < y. But if z : x < z : y, then x > y

V.11. Transitivity of equal ratios. If u : v = w : x and w : x = y : z, then u : v = y : z.

V.12. If x₁:y₁ = x₂:y₂ = … = x_n : y_n, then each of these ratios also equals the ratio (x₁ + x₂ + … + x_n) : (y₁ + y₂ + … + y_n).

V.13. Substitution of equal ratios in inequalities of ratios. If u : v = w : x and w : x > y : z, then u : v > y : z.

V.14. If w : x = y : z and w > y, then x > z.

V.15. x : y = nx : ny.

V.16. Alternate proportions. If w : x = y : z, then w : y = x : z.

V.17. Proportional taken jointly implies proportional taken separately. If (w + x):x = (y + z):z, then w : x = y : z.

V.18. Proportional taken separately implies proportional taken jointly. (A converse to V.17.) If w : x = y : z, then (w + x):x = (y + z):z.

V.19. If (w + x) : (y + z) = w : y, then (w + x) : (y + z) = x : z, too.

V.19.Cor. Proportions in conversion. If (u + v) : (x + y) = v : y, then (u + v) : (x + y) = u : x.

V.20 is just a preliminary proposition to V.22, and V.21 is just a preliminary proposition to V.23.

V.22. Ratios ex aequali. If x₁:x₂ = y₁:y₂, x₂:x₃ = y₂:y₃, … , and x_n-1:x_n = y_n-1:y_n, then x₁:x_n = y₁:y_n.

V.23. Perturbed ratios ex aequali. If u : v = y : z and v : w = x : y, then u : w = x : z.

V.24. If u : v = w : x and y : v = z : x, then (u + y):v = (w + z):x.

V.25. If w : x = y : z and w is the greatest of the four magnitudes while z is the least, then w + z > x + y.

Logical structure of Book V

Book V is on the foundations of ratios and proportions and in no way depends on any of the previous Books. Book VI contains the propositions on plane geometry that depend on ratios, and the proofs there frequently depend on the results in Book V. Also Book X on irrational lines and the books on solid geometry, XI through XIII, discuss ratios and depend on Book V. The books on number theory, VII through IX, do not directly depend on Book V since there is a different definition for ratios of numbers.Although Euclid is fairly careful to prove the results on ratios that he uses later, there are some that he didn’t notice he used, for instance, the law of trichotomy for ratios. These are described in the Guides to definitions V.Def.4 through V.Def.7.

* Some of the propositions in Book V require treating definition V.Def.4 as an axiom of comparison. One side of the law of trichotomy for ratios depends on it as well as propositions 8, 9, 14, 16, 21, 23, and 25. Some of Euclid’s proofs of the remaining propositions rely on these propositions, but alternate proofs that don’t depend on an axiom of comparison can be given for them.

Propositions 1, 2, 7, 11, and 13 are proved without invoking other propositions. There are moderately long chains of deductions, but not so long as those in Book I.

The first six propositions excepting 4 have to do with arithmetic of magnitudes and build on the Common Notions. The next group of propositions, 4 and 7 through 15, use the earlier propositions and definitions 4 through 7 to develop the more basic properties of ratios. And the last 10 propositions depend on most of the preceding ones to develop advanced properties.

2018년 12월 20일2018년 12월 20일

4권

Definition 1.: A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.; 한 다각형의 모든 각(꼭짓점)들이 다른 다각형의 변들에 놓여 있으면 그 다각형은 내접하고 있다.
Definition 2.: Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.; 한 다각형의 변들이 다른 다각형의 각들을 지나면, 그 다각형은 다른 다각형에 외접하고 있다.
Definition 3.: A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.; 한 다각형의 모든 각들이 원둘레에 놓여 있으면, 그 다각형은 원에 내접하고 있다.
Definition 4.: A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.; 한 다각형의 모든 변들이 원둘레와 접하고 있으면, 그 다각형은 원에 외접하고 있다.
Definition 5.: Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.; 어떤 원의 둘레가 어떤 다각형의 모든 변들과 접하고 있으면, 그 원은 그 다각형에 내접하고 있다.
Definition 6.: A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.; 어떤 원의 둘레가 어떤 다각형의 모든 각을 지나면, 그 원은 그 다각형에 외접하고 있다.
Definition 7.: A straight line is said to be fitted into a circle when its ends are on the circumference of the circle.; 직선의 양 끝점이 원둘레에 놓여 있으면, 그 직선은 원에 걸쳐 있다.

Propositions

Proposition 1.: To fit into a given circle a straight line equal to a given straight line which is not greater than the diameter of the circle.

어떤 원과, 그 원의 지름보다 짧은 직선을 주었을 때, 그 원에 주어진 직선과 같은 길이인 직선을 걸쳐 놓아라.

Proposition 2.

To inscribe in a given circle a triangle equiangular with a given triangle.

주어진 삼각형과 각이 모두 같은 삼각형을 주어진 원에 내접시켜라.

Proposition 3.

To circumscribe about a given circle a triangle equiangular with a given triangle.

주어진 삼각형과 각이 모두 같은 삼각형을 주어진 원에 외접시켜라.

Proposition 4.

To inscribe a circle in a given triangle.

주어진 삼각형에 원을 내접시켜라.

Proposition 5.

To circumscribe a circle about a given triangle.

주어진 삼각형에 원을 외접시켜라. Corollary. When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled.

이때, 원의 중심이 삼각형 안에 놓이면 예각삼각형, 변에 놓이면 직각삼각형, 바깥에 놓이면 둔각삼각형이다.

Proposition 6.

To inscribe a square in a given circle.

주어진 원에 정사각형을 내접시켜라.

Proposition 7.

To circumscribe a square about a given circle.

주어진 원에 정사각형을 외접시켜라.

Proposition 8.

To inscribe a circle in a given square.

주어진 정사각형에 원을 내접시켜라.

Proposition 9.

To circumscribe a circle about a given square.

주어진 정사각형에 원을 외접시켜라.

Proposition 10.

To construct an isosceles triangle having each of the angles at the base double the remaining one.

두 밑각의 크기가 나머지 한 각의 크기의 두 배가 되는 이등변삼각형을 그려라.

Proposition 11.

To inscribe an equilateral and equiangular pentagon in a given circle.

주어진 원에 정오각형을 내접시켜라.

Proposition 12.

To circumscribe an equilateral and equiangular pentagon about a given circle.

주어진 원에 정오각형을 외접시켜라.

Proposition 13.

To inscribe a circle in a given equilateral and equiangular pentagon.

정오각형에 원을 내접시켜라.

Proposition 14.

To circumscribe a circle about a given equilateral and equiangular pentagon.

정오각형에 원을 외접시켜라.

Proposition 15.

To inscribe an equilateral and equiangular hexagon in a given circle.

주어진 원에 정육각형을 내접시켜라.

Corollary. The side of the hexagon equals the radius of the circle.

And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon.

And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it.

정육각형의 변은 원의 반지름과 같다.

오각형의 경우처럼, 원둘레에 놓이는 점들에서 원에 접하는 직선을 그으면, 정육각형을 원에 외접하도록 그릴 수 있다. 오각형의 경우와 마찬가지다.

오각형의 경우와 마찬가지로 정육각형을 주었을 때 내접하는 원과 외접하는 원을 그릴 수 있다.

Proposition 16.: To inscribe an equilateral and equiangular fifteen-angled figure in a given circle.; 주어진 원에 정십오각형을 내접시켜라.
Corollary. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular.And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it.; 오각형의 경우와 마찬가지로, 원둘에에 놓이는 점들에서 원에 접하는 직선을 기으면, 정십오각형을 이 원에 외접하도록 그릴 수 있다.; 오각형의 경우에 증명한 것과 마찬가지로 정십오각형을 주었을 때 내접하는 원과 외접하는 원을 그릴 수 있다.

Guide to Book IV

All but two of the propositions in this book are constructions to inscribe or circumscribe figures.

Figure	Inscribe figure in circle	Circumscribe figure about circle	Inscribe circle in figure	Circumscribe circle about figure
Triangle	IV.2	IV.3	IV.4	IV.5
Square	IV.6	IV.7	IV.8	IV.9
Regular pentagon	IV.11	IV.12	IV.13	IV.14
Regular hexagon	IV.15	IV.15,Cor	IV.15,Cor	IV.15,Cor
Regular 15-gon	IV.16	IV.16,Cor	IV.16,Cor	IV.16,Cor

There are only two other propositions. Proposition IV.1 is a basic construction to fit a line in a circle, and proposition IV.10 constructs a particular triangle needed in the construction of a regular pentagon.

Logical structure of Book IV

The proofs of the propositions in Book IV rely heavily on the propositions in Books I and III. Only one proposition from Book II is used and that is the construction in II.11 used in proposition IV.10 to construct a particular triangle needed in the construction of a regular pentagon.

Most of the propositions of Book IV are logically independent of each other. There is a short chain of deductions, however, involving the construction of regular pentagons.

Dependencies within Book IV
1, 5	10
2, 10	11
11	12
1, 2, 11	16

2018년 12월 20일2018년 12월 20일

3권

Definition 1.: Equal circles are those whose diameters are equal, or whose radii are equal.; 지름이나 반지름이 같은 원은 같은 원이다.
Definition 2.: A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.; 직선이 원과 만나지만 아무리 길게 늘여도 원을 자르지 않을 때 접한다고 한다.
Definition 3.: Circles are said to touch one another which meet one another but do not cut one another.; 원이 서로 만나지만 서로 자르지 않을 때 접한다고 한다.
Definition 4.: Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal.; 원의 중심에서 직선에 내린 수선의 길이가 같을 때 그 직선들은 중심에서 같은 거리에 있다고 말한다.
Definition 5.: And that straight line is said to be at a greater distance on which the greater perpendicular falls.; 중심에서 직선에 내린 수선의 길이가 더 길면 더 멀리 있다고 말한다.
Definition 6.: A segment of a circle is the figure contained by a straight line and a circumference of a circle.; 활꼴은 원둘레와 직선으로 이루어진 도형이다.
Definition 7.: An angle of a segment is that contained by a straight line and a circumference of a circle.활꼴의 각은 원둘레와 직선이 이루는 각이다.
Definition 8.: An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined.활꼴의 내부 원주각이란 활꼴을 이루는 원둘레의 한 점에서 활꼴의 밑변을 이루는 직선의 양 끝점으로 두 직선을 그었을 때, 그 두 직선이 만드는 각이다.
Definition 9.: And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.각을 만드는 두 직선이 원을 자를 때, 그 각은 원둘레에 서 있다고 말한다.
Definition 10.: A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.부채꼴이란 원의 중심에서 어떤 각을 만들었을 때, 그 각을 만드는 두 직선과 그 직선들에 의해 잘린 원둘레로 둘러싸인 도형이다.
Definition 11.: Similar segments of circles are those which admit equal angles, or in which the angles equal one another.; 각들의 크기가 같은 활꼴은 닮은 활꼴이다.

Propositions

Proposition 1.

To find the center of a given circle.

주어진 원의 중심을 찾아라.

Corollary. If in a circle a straight line cuts a straight line into two equal parts and at right angles, then the center of the circle lies on the cutting straight line.

원 안에서 한 직선이 한 직선을 수직이등분하면 중심은 그 수직이등분선 위에 있다.

Proposition 2.

If two points are taken at random on the circumference of a circle, then the straight line joining the points falls within the circle.

원 둘에 위에 임의의 두 점이 주어졌을 때 두 점을 잇는 선분은 원 안에 있다.

Proposition 3.

If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it.

중심을 지나는 직선이 중심을 지나지 않는 현을 이등분한다면 그 직선은 현을 수직으로 자른다. 수직으로 자른다면 이 선은 이등분선이다.

Proposition 4.

If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another.

원에서 중심을 지나지 않는 두 현이 서로 서로 자른다면 현들은 서로를 이등분하지 않는다.

Proposition 5.

If two circles cut one another, then they do not have the same center.

두 원이 서로를 자른다면 같은 중심을 가지지 않는다.

Proposition 6.

If two circles touch one another, then they do not have the same center.

두 원이 서로 접한다면 같은 중심을 가지지 않는다.

Proposition 7.

If on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines fall from the point on the circle, one on each side of the least straight line.

원의 지름 위에 중심이 아닌 점을 잡고 그 점에서 원둘레로 직선을 그으면 중심을 지나는 것이 가장 길고 지름에서 그 직선을 빼고 남는 것이 가장 짧다. 다른 직선들도 원의 중심에 가까운 것이 먼 것보다 더 길다. 길이가 같은 직선은 둘씩 존재하며, 가장 짧은 직선의 양쪽에 하나씩 존재한다.

Proposition 8.

If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least.원의 바깥에 한 점에서 원둘레로 직선을 그어 하나는 중심을 지나도록 긋고 다른 직선들은 적당히 긋는다면 원의 오목한 부분에 닿는 직선 가운데 중심을 지나는 것이 가장 길다. 다른 직선들은 중심을 지나는 직선에 가까울수록 멀리 있는 것보다 더 길다. 반대로 원둘레의 볼록한 부분에 닿는 직선들 가운데에는 그 점과 지름 사이에 놓이는 직선이 가장 짧다. 다른 직선들은 가장 짧은 직선에 가까울수록 멀리 있는 것보다 더 짧다. 길이가 같은 직선은 그 점에서 둘씩 그을 수 있으며, 가장 짧은 직선의 양쪽에 하나씩 존재한다.

Proposition 9.

If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle.원의 한 점에서 원 둘레로 직선을 그을 때 길이가 같은 직선을 둘보다 더 많이 그을 수 있다면 그 점은 원의 중심이다.

Proposition 10.

A circle does not cut a circle at more than two points.원은 다른 원을 두 점보다 많은 점에서 자를 수 없다.

Proposition 11.

If two circles touch one another internally, and their centers are taken, then the straight line joining their centers, being produced, falls on the point of contact of the circles.두 원이 내접할 때, 두 원의 중심을 잇는 직선을 늘리면 접점에 닿는다.

Proposition 12.

If two circles touch one another externally, then the straight line joining their centers passes through the point of contact.두 원이 외접할 때, 두 원의 중심을 잇는 직선은 접점을 지난다.

Proposition 13.

A circle does not touch another circle at more than one point whether it touches it internally or externally..원은 내접하거나 외접하거나 한 점보다 많은 점에서 접할 수 없다.

Proposition 14.

Equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another.원에 길이가 같은 현은 중심에서 같은 거리에 있고 중심에서 같은 거리에 있는 현은 길이가 같다.

Proposition 15.

Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.지름은 가장 긴 현이고 나머지 현은 중심에 가까울수록 먼 것보다 더 길다.

Proposition 16.

The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.

지름의 끝점에서 지름에 수직이 되도록 직선을 그으면 그 직선은 원의 바깥에 있다. 그 직선과 원둘레 사이에는 어떤 직선도 놓일 수 없다. 반원이 만드는 각은 어떤 직선 예각보다도 크고 그것을 뺐을 때 남는 각은 어떤 직선 예각보다 작다.

Corollary. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its end touches the circle.

지름의 끝점에서 지름에 수직인 직선은 접선이다.

Proposition 17.

From a given point to draw a straight line touching a given circle.

어떤 점과 원을 주었을 때, 원에 접하도록 그 점에서 직선을 그어라.

Proposition 18.

If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.어떤 직선이 원에 접한다면 접점과 원의 중심을 지나는 직선은 접선과 수직이다.

Proposition 19.

If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.어떤 직선이 원에 접한다면 접점에서 그 직선에 수직인 직선을 그으면 원의 중심을 지난다.

Proposition 20.

In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.원의 중심에서 만든 각(중심각)은 둘레에서 만든 각(원주각)이 같은 호를 밑변으로 가지면, 중심각 크기는 원주각 크기의 두 배가 된다.

Proposition 21.

In a circle the angles in the same segment equal one another.같은 활꼴의 원주각은 모두 같다.

Proposition 22.

The sum of the opposite angles of quadrilaterals in circles equals two right angles.원에 내접하는 사각형은 마주보는 두 각의 합이 직각 둘을 더한 것과 같다.

Proposition 23.

On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.어떤 현에서 닮은꼴이면서 서로 다른 활꼴을 같은 쪽에 그릴 수 없다.

Proposition 24.

Similar segments of circles on equal straight lines equal one another.

길이가 같은 현애 대해 닮은꼴 활꼴을 만들면 그 활꼴들은 서로 같다.

Proposition 25.

Given a segment of a circle, to describe the complete circle of which it is a segment.

어떤 활꼴이 있을 때, 그 활꼴을 가지는 완전한 원을 그려라.

Proposition 26.

In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.크기가 같은 원에서 크기가 같은 중심각 또는 원주각들은 같은 호에 대응한다.

Proposition 27.

In equal circles angles standing on equal circumferences equal one another whether they stand at the centers or at the circumferences.크기가 같은 원에서 길이가 같은 호들에 대응하는 중심각과 원주각은 서로 같다.

Proposition 28.

In equal circles equal straight lines cut off equal circumferences, the greater circumference equals the greater and the less equals the less.크기가 같은 원들에서 길이가 같은 직선들은 같은 호를 만든다. 긴 호는 긴 호와 짧은 호는 짧은 호와 같다.

Proposition 29.

In equal circles straight lines that cut off equal circumferences are equal.

크기가 같은 원들에 있는 길이가 같은 호들은 같은 길이인 직선에 대응한다.

Proposition 30.

To bisect a given circumference.

주어진 호를 이등분하여라.

Proposition 31.

In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle.반원의 내부 원주각은 직각이다. 반원보다 더 큰 활꼴의 내부 원주각은 직각보다 작고 반원보다 작은 활꼴의 내부 원주각은 직각보다 크다. 반원보다 더 큰 활꼴의 각은 직각보다 크고 반원보다 더 작은 활꼴의 각은 직각보다 작다.

Proposition 32.

If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle.

직선이 원에 접하고 접점에서 직선을 그어서 원을 자르게 한다면 그 직선과 접선이 만드는 각은 반대쪽 활꼴의 내부 원주각과 크기가 같다.

Proposition 33.

On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle.어떤 직선과 각을 주었을 때, 그 직선에 활꼴을 그려서 그 활꼴의 내부 원주각이 주어진 각과 크기가 같게 만들어라.

Proposition 34.

From a given circle to cut off a segment admitting an angle equal to a given rectilinear angle. 어떤 원과 각을 주었을 떄, 내부 원주각이 주어진 각과 같은 크기가 되는 활꼴을 잘라내라.

Proposition 35.

If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other.원에서 두 직선이 서로 자르고 지나가면 한 직선의 토막들을 가지고 만든 직사각형은 다른 직선의 토막들로 만든 직사각형과 넓이가 같다.

Proposition 36.

If a point is taken outside a circle and two straight lines fall from it on the circle, and if one of them cuts the circle and the other touches it, then the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the tangent.원 밖에 있는 점에서 원으로 두 직선을 그어 한 직선은 원을 자르게 하고 다른 직선은 접하게 한다면 원을 자르른 직선의 전체 길이과그 직선에서 원의 볼록한 둘레에 닿기까지의 길이를 가지고 만든 직사각형은 접을 가지고 만든 정사각형과 넓이가 같다.

Proposition 37.

If a point is taken outside a circle and from the point there fall on the circle two straight lines, if one of them cuts the circle, and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference equals the square on the straight line which falls on the circle, then the straight line which falls on it touches the circle.

원의 바깥에 있는 점에서 원으로 두 직선을 그어서 한 직선은 원을 자르게 하고 다른 한 직선은 원에 닿도록 했을 때, 원을 자르는 직선의 전체 길이와 그 직선에서 원의 볼록한 둘레에 닿기까지 길이로 만든 직사각형이 원에 닿는 직선으로 만든 정사각형과 넓이가 같다면 원에 닿는 직선은 원에 접한다.

2018년 12월 20일2018년 12월 20일

2권

Definition 1. Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle. 직사각형은 직각을 끼고 있는 두 변으로 만들었다고 한다.

Definition 2 And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon. 평행사변형에서 맞모금으로 만든 한 평행사변형과 남은 부분 둘을 더해서 만든 도형을 기역 모양 도형(gnomon:carpenter’s square) 으로 부른다.

Propositions

Proposition 1. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.

직선이 둘 있을 때 하나를 임의의 숫자로 자르면 두 직선으로 만든 직사각형 넓이는 자르지 않은 직선과 잘려진 토막으로 만든 직선들로 만든 직사각형 넓이를 모두 더한 것과 같다.

Proposition 2. If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole.

어떤 직선을 아무 점이나 잡아서 토막 냈다면 원래 직선과 토막들을 가지고 만든 직사각형들 넓이를 더하면 원래 직선으로 만든 정사각형 넓이와 같다.

Proposition 3. If a straight line is cut at random, then the rectangle contained by the whole and one of the segments equals the sum of the rectangle contained by the segments and the square on the aforesaid segment.

어떤 직선을 아무 점이나 잡아서 자르면 그 가운데 한 토막과 전체 직선으로 만든 직사각형 넓이는 그 한 토막으로 만든 정사각형과 토막들을 가지고 만든 직사각형의 넓이를 더한 것과 같다.$a(a+b)=a^2 +ab$

Proposition 4. If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments.

어떤 직선을 아무 점이나 잡아서 토막냈다면 전체 직선으로 만든 정사각형 넓이는 각각의 토막들로 만든 정사각형 넓이에 토막들로 만든 직사각형 넓이의 두 배를 더한 것과 같다.$(a+b)^2 =a^2 +b^2 +2ab$

Proposition 5. If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.

직선을 길이가 같게 둘로 자르고 길이가 다르게 둘로 잘랐을 때, 길이가 다른 두 토막으로 만든 직사각형에 자른 점들 사이의 직선으로 만든 정사각형을 더하면 전체 길이의 절반으로 만든 정사각형과 넓이가 같다.

Proposition 6. If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line.

어떤 직선을 이등분한 다음 다른 어떤 직선을 한 줄이 되도록 이어 붙이면 직선 전체와 붙인 직선으로 만든 직사가형에 반토막으로 만든 정사각형을 더하면 반토막에다 직선을 붙인 것으로 만든 정사각형과 넓이가 같다.

Proposition 7. If a straight line is cut at random, then the sum of the square on the whole and that on one of the segments equals twice the rectangle contained by the whole and the said segment plus the square on the remaining segment.

직선에 임으의 한 점을 잡아서 직선을 자르면 전체로 만든 정사각형에 한 토막을 만든 정사각형을 더하면 직선 전체와 그 한 모막을 만든 직사각형 넓이 두 배에 다른 토막으로 만든 정사각형 넓이를 더한 것과 같다.

Proposition 8. If a straight line is cut at random, then four times the rectangle contained by the whole and one of the segments plus the square on the remaining segment equals the square described on the whole and the aforesaid segment as on one straight line.

직선에 아무 점이라도 좋으니 잡아서 직선을 자르면 직선 전체와 한 토막으로 만든 직사각형의 네 배에다 다른 토막으로 만든 정사각형을 더하면 직선 전체에다 그 한 토막을 더한 직선으로 만든 정사각형과 넓이가 같다.

Proposition 9. If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section.

직선을 길이가 같게 두 토막을 내고 또 다른 점을 잡아 길이가 다르게 두 토막을 낸다면 길이가 다른 토막들로 만든 정사각형 둘을 더하면 그것은 전체 길이의 절막을 가지고 만든 정사각형에 자른 점들 사이의 직선을 가지고 만든 정사각형을 더한 것을 두 배 한 것과 넓이가 같다.

Proposition 10. If a straight line is bisected, and a straight line is added to it in a straight line, then the square on the whole with the added straight line and the square on the added straight line both together are double the sum of the square on the half and the square described on the straight line made up of the half and the added straight line as on one straight line.

어떤 직선을 이등분하고 거기에 다른 어떤 직선을 한 줄이 되도록 붙이면 원래 직선에 다른 어떤 직선을 붙인 것으로 정사각형을 만들고 거기에 붙인 직선으로 만든 정사각형을 더하면 그것은 반토막으로 만든 정사각형과 반토막에다 다른 어떤 직선을 붙인 것을 가지고 만든 정사각형을 더한 것의 두 배가 된다.

Proposition 11. To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. 주어진 직선을 적당히 잘라서 전체 길이와 한 토막으로 만든 직사각형이 다른 토막으로 만든 정사각형과 넓이가 같도록 만드시오. Proposition 12. In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.

둔각삼각형에서 둔각의 대변 위의 정사각형은 둔각을 끼는 두 변 위의 정사각형의 합보다, 둔각을 끼는 변의 하나와 이 변에 수선이 내려지고, 이 둔각에의 수선에 의해서 외부에 잘려진 선분으로 에워싸인 직사각형의 2배만큼 크다.

Proposition 13. In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle.

예각삼각형에서 예각의 대변 위의 정사각형은 예각을 끼는 두 변 위의 정사각형의 합보다, 예각을 끼는 변의 하나와 이 변에 수선이 내려지고, 이 예각에의 수선에 의해서 외부에 잘려진 선분으로 에워싸인 직사각형의 2배만큼 작다.

Proposition 14. To construct a square equal to a given rectilinear figure. 주어진 다각형과 넓이가 같은 정사각형을 만들어라.

Guide to Book II

The subject matter of Book II is usually called “geometric algebra.” The first ten propositions of Book II can be easily interpreted in modern algebraic notation. Of course, in doing so the geometric flavor of the propositions is lost. Nonetheless, restating them algebraically can aid in understanding them. The equations are all quadratic equations since the geometry is plane geometry.

II.1. If y = y₁ + y₂ + … + y_n, then xy = x y₁ + x y₂ + … + x y_n. This can be stated in a single identity as

x (y₁ + y₂ + … + y_n) = x y₁ + x y₂ + … + x y_n.

II.2. If x = y + z, then x² = xy + xz. This can be stated in various ways in an identity of two variables. For instance,

(y + z)² = (y + z) y + (y + z) z,

x² = xy + x (x – y).

II.3. If x = y + z, then xy = yz + y². Equivalent identities are

(y + z)y = yz + y²,

and

xy = y(x – y) + y².

II.4. If x = y + z, then x² = y² + z² + 2yz. As an identity,

(y + z)² = y² + z² + 2yz.

II.5 and II.6. (y + z) (y – z) + z² = y².

II.7. if x = y + z, then x² + z² = 2xz + y². As an identity,

x² + z² = 2xz + (x – z)².

II.8. If x = y + z, then 4xy + z² = (x + y)². As an identity,

4xy + (x – y)² = (x + y)².

II.9 and II.10. (y + z)² + (y – z)² = 2 (y² + z²).

The remaining four propositions are of a slightly different nature. Proposition II.11 cuts a line into two parts which solves the equation a (a – x) = x² geometrically. Propositions II.12 and II.13 are recognizable as geometric forms of the law of cosines which is a generalization of I.47. The last proposition II.14 constructs a square equal to a given rectilinear figure thereby completeing the theory of areas begun in Book I.

Logical structure of Book II

The proofs of the propositions in Book II heavily rely on the propositions in Book I involving right angles and parallel lines, but few others. For instance, the important congruence theorems for triangles, namely I.4, I.8, and I.26, are not invoked even once. This is understandable considering Book II is mostly algebra interpreted in the theory of geometry.

The first ten propositions in Book II were written to be logically independent, but they could have easily been written in logical chains which, perhaps, would have shortened the exposition a little. The remaining four propositions each depend on one of the first ten.

2018년 12월 19일2018년 12월 23일

Definitions 정의Definition 1.

A point is that which has no part.; 점은 부분이 없는 것이다. 다르게 말하면 더는 쪼갤 수 없다는 것이다. 왜 이렇게 부분이 없다고 정의했을까? 점에서 선이나 면으로 나간 것이 아니라 면이나 선에서 시작해서 점을 추상화하고 다시 선과 면을 추상해 나간 것이 아닐까 생각한다. 아무튼 점은 길이나 폭이 없고 그저 위치만이 있을 뿐이다.
Definition 2.: A line is breadthless length.; 선은 폭이 없는 길이이다. 선은 점이 움직여 간 것으로 생각하면 된다. 물론 선이 가지고 있는 완비성(completeness)까지 도달하지는 못했지만 직관이 아닌 추상으로 선을 정의한 점은 높이 사야 한다. 다음에 직선을 따로 정의하고 있으므로 선(line)은 곡선(curve)을 포함하고 있다.
Definition 3.: The ends of a line are points.; 선은 점으로 끝난다. 선을 자르면 그 맨끝에는 점들이 있다는 말이다.
Definition 4.: A straight line is a line which lies evenly with the points on itself.; 직선은 점들이 한결같이 고르게 놓인 것이다. 직선은 울퉁불퉁하지 않은 곧은 선이라는 말이다. 오늘날은 직선을 line으로 곡선은 curve로 부르고 있다.
Definition 5.: A surface is that which has length and breadth only.; 면은 길이와 폭 만을 가진 것이다. 선과 마찬가지로 면은 선이 옆으로 움직여간 것이라고 생각하면 좋겠다.
Definition 6.: The edges of a surface are lines.; 면의 모서리는 선이다. 선이 점으로 끝나는 것과 마찬가지다. 면을 자르면 맨끝에는 선들이 있다.
Definition 7.: A plane surface is a surface which lies evenly with the straight lines on itself.; 평면은 직선이 고르게 펼쳐진 것이다.
Definition 8.: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.; 같은 평면에 놓인 서로 다른 두 선이 한 점에서 만날 때, 두 선 서로에 대한 기울기가 평면각이다.; 바로 이어지는 9번에 직선각을 따로 언급하였으므로 평면각은 곡선을 변으로 가질 수도 있다. 오늘날 각에 대한 정의보다 넓은 의미를 품고 있다. 곡선이 이루는 각은 접선이 이루는 각으로 정의한다.
Definition 9.: And when the lines containing the angle are straight, the angle is called rectilinear.; 각을 품은 선들이 직선일 때, 각은 직선각(직선으로 싸인 각)이라고 한다.
Definition 10.: When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.; 직선에 서 있는 어떤 직선이 이웃한 각이 서로 같을 때, 두 각은 직각이고 서 있는 직선은 다른 직선의 수선이다.
Definition 11.: An obtuse angle is an angle greater than a right angle.; 직각보다 큰 각은 둔각(무딘각)이다.
Definition 12.: An acute angle is an angle less than a right angle.; 직각보다 작은 각은 예각(뾰족각)이다.
Definition 13.: A boundary is that which is an extremity of anything.; 경계는 어떤 것의 끝단이다.
Definition 14.: A figure is that which is contained by any boundary or boundaries.; 도형은 경계 또는 경계들을 포함한 것이다.
Definition 15.: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.; 원은 한점에서 도형까지 이은 직선이 모두 길이가 같은 하나의 선을 품은 평면 도형이다. 한 정점에서 거리가 일정한 점들의 집합이 바로 원이다.
Definition 16.: And the point is called the center of the circle.; 바로 그 점이 원의 중심이다.
Definition 17.: A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.; 원의 지름은 중심을 지나 양쪽으로 원둘레에 다다를 때까지 그린 직선이다. 지름은 원을 이등분한다.
Definition 18.: A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.; 반원은 지름과 지름으로 잘린 원주를 포함한 도형이다. 반원과 원은 중심이 같다.
Definition 19.: Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.; 직선 도형(변형)은 직선들로 이루어졌다. 셋이 있는 삼변형, 넷이 있는 사변형, 넷보다 많은 다변형이 있다. 요즘 말로하면 다각형이라고 옮기면 된다.
Definition 20.: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.; 삼변형에서 세변이 모두 같으면 정삼각형 두변이 같으면 이등변 삼각형 셋이 모두 다른 것은 부등변 삼각형이다.
Definition 21.: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.; 직각이 있으면 직각삼각형 둔각이 있으면 둔각삼각형 세각이 모두 예각이면 예각삼각형이다.
Definition 22.: Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.; 사변형에서 네변이 같고 각이 모두 직각이면 정사각형, 네각은 직각이지만 변이 같지 않으면 직사각형, 네변은 같지만 각이 다르면 마름모, 마주보는 각이 같지만 직각이 아니면 평행사변형, 모두 다르면 부등변사변형이다.
Definition 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.; 평행선은 양쪽으로 끝없이 늘려도 어떤 쪽에서도 서로 만나지 않는 같은 평면에 있는 두 직선이다.

Postulates 공준

Let the following be postulated:공준과 아래 공통 관념을 모두 오늘날은 공리(Axiom)라고 부른다.

Postulate 1.: To draw a straight line from any point to any point.; 임의의 점에서 임의의 점으로 직선을 그릴 수 있다.; 임의의 두 점이 있으면 그 두 점을 끝점으로 하는 선분 하나를 그릴 수 있다. 선분 위 점들은 모두 끝점 사이에 있다고 생각한다.
Postulate 2.: To produce a finite straight line continuously in a straight line.; 선분을 이어서 직선을 만들 수 있다.; 선분을 한 없이 늘여서 직선을 만들 수 있음을 말한다.
Postulate 3.: To describe a circle with any center and radius.; 임의의 중심과 반지름을 가진 원을 그릴 수 있다.; 이 공리는 원을 그리기 위해 반지름을 나타내는 선분을 움직여도 선분 길이가 달라지지 않도록 공간상의 거리가 정의되었음을 함축한다.
Postulate 4.: That all right angles equal one another.; 모든 직각은 서로 같다.; 직각은 직선이 다른 직선과 만날 때 교점에서 바로 옆에 있는 각이 서로 같을 때 만들어지는 각이다. 한 직선이 다른 직선과 수직으로 만날 때 교차각이 모두 90도인 것이다. 그러나 정의만 보면 반드시 그렇다고 할 수 없다. 심지어 정의에는 각이 항상 같은 수가 된다는 것조차 명시되어 있지 않다. 두 직선이 어떤 특정 지점에서 만나면 교차각이 90도이고, 다른 지점에서 만나면 다른 값이 되는 세상을 상상할 수 있다. 그러므로 모든 직각은 같다는 공리는 이런 일이 벌어지지 않음을 단언하는 것이다. 다시 말해 이 공리는 직선이 어느 부분에서나 같은 모양이라는 것, 즉 곧음(straightness) 조건을 말하고 있는 것이다.
Postulate 5.: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.; 두 직선과 한 직선이 만날 때 있는 두 직선을 한없이 늘리면 같은 쪽에 있는 내각을 더해서 직각 둘(180도)보다 작은 쪽에서 만난다.
다른 공준들은 간단하지만 이 공준은 좀 길다. 훗날 이 다섯째 공준이 혹시 다른 공준들로 증명할 수 있는 정리가 아닐까 생각한 이들이 있었다. 증명에 나섰던 이들이 증명은 찾지 못하고 찾아낸 기하학이 바로 비-유클리드 기하학(Non-Euclidean geometry)이다. 이 공준은 ‘한 직선 위에 있지 않은 한 점을 지나는 직선은 단 하나뿐이다.’라는 공리와 같은 공준인데 이 때 단 하나가 아니라 ‘없다’거나 ‘무수히 많다’로 바꾸어도 새로운 공리계를 이룬다는 걸 알아냈다. 차례로 타원 기하학과 쌍곡선 기하학이다.

Common Notions 공통 관념

Common notion 1.: Things which equal the same thing also equal one another.; 똑같은 것과 같은 것들은 서로 같다.
Common notion 2.: If equals are added to equals, then the wholes are equal.; 같은 것에 같은 것을 더하면 그 더한 전체는 여전히 같다.
Common notion 3.: If equals are subtracted from equals, then the remainders are equal.; 같은 것에서 같은 것을 덜어내어도 그 나머지들은 여전히 같다.
Common notion 4.: Things which coincide with one another equal one another.; 포개어서 같은 것들은 서로 같다. 다시 말해 평행이동이나 대칭이동하여 완전히 포개지는 것들은 서로 같은 것이다.
Common notion 5.: The whole is greater than the part.; 전체는 부분보다 크다.

공통 관념은 한종류의 양(magnitude)에 대한 것이다. 원론에는 직선, 각, 평면도형, 입체가 가지는 다양한 양이 있다. 길이, 각의 크기, 넓이, 부피가 그것이다. 다른 종류의 양은 서로 같다고 하거나 서로 더할 수 없다. 길이를 직사각형 넓이에 더하거나 하는 일은 생각하지 않는다. 삼각형 넓이와 직사각형 넓이가 같고 직사각형 넓이와 정사각형 넓이가 같다면 삼각형과 정사각형 넓이가 같다고 말한다는 것이다. 5번은 어떤 양 $B$가 $A$의 부분이라면 다른 나머지 양 $C$가 있다는 것이다. 달리 표현하면 $A>B$이면 $A=B+C$를 만족하는 $C$가 존재한다 것이다.

Propositions 명제

Proposition 1.: To construct an equilateral triangle on a given finite straight line.; 주어진 선분을 변으로 하는 정삼각형을 만들 수 있다.

Proposition 2.

To place a straight line equal to a given straight line with one end at a given point.; 주어진 선분과 같은 선분을 주어진 한 점을 끝점으로 하여 만들 수 있다.
Proposition 3.: To cut off from the greater of two given unequal straight lines a straight line equal to the less.; 서로 다른 선분이 있을 때 작은 것과 같은 선분으로 큰 것을 자를 수 있다.
Proposition 4.: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.; 두 삼각형에서 두변과 사이에 낀 각이 서로 같다면 나머지 변과 각들은 서로 같다. 다시 말하면 합동(congruence)이다.
Proposition 5.: In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.; 이등변 삼각형에서 두 밑각은 서로 같다. 두 밑각 아래에 있는 외각도 서로 같다.
Proposition 6.: If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.; 삼각형에서 두 각이 서로 같다면 마주보는 변들이 서로 같다.
Proposition 7.: Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end.; 선분의 양 끝 점 시작해서 한 점에서 만나는 두 직선을 그었다면, 같은 선분의 양 끝 점에서 주어진 두 직선과 각각 같은 직선으로 같은 쪽에 있는 다른 점에서 만나게 할 수는 없다.
Proposition 8.: If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.; 두 삼각형이 두 변이 서로 같고 밑변도 서로 같다면 각도 서로 같다.
Proposition 9.: To bisect a given rectilinear angle.; 직선이 만나서 이루는 각을 이등분 할 수 있다.
Proposition 10.: To bisect a given finite straight line.; 선분의 이등분을 할 수 있다.
Proposition 11.: To draw a straight line at right angles to a given straight line from a given point on it.; 직선 위에 주어진 점에서 수직인 직선을 그을 수 있다.
Proposition 12.: To draw a straight line perpendicular to a given infinite straight line from a given point not on it.; 직선과 그 위에 있지 않은 한 점이 주어지면 그 점을 지나고 직선에 수직인 직선을 그을 수 있다.
Proposition 13.: If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.; 두 직선이 만나면 이직각(180도)을 이루거나 두 각의 합이 이직각을 이룬다.
Proposition 14.: If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.; 어떤 직선 위에 있는 한 점에서 같은 쪽에 있지 않고 이루는 각의 합이 이직각인 두 직선이 있다면 두 직선은 서로를 품고 있다.
Proposition 15.: If two straight lines cut one another, then they make the vertical angles equal to one another.; 두 직선이 서로를 자르면 마주보는 각(맞꼭지각)은 서로 같다.
Corollary. If two straight lines cut one another, then they will make the angles at the point of section equal to four right angles.; 두 직선이 서로를 자르면 직각 넷으로 나누어진 부분으로 만들 수 있다.
Proposition 16.: In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.; 어떤 삼각형이든 변이 있다면 외각은 건너편 내각보다 크다.
Proposition 17.: In any triangle the sum of any two angles is less than two right angles.; 어떤 삼각형이든 두 각의 합은 180보다 작다.
Proposition 18.: In any triangle the angle opposite the greater side is greater.; 어떤 삼각형이든 마주보는 변이 큰 각이 더 크다.
Proposition 19.: In any triangle the side opposite the greater angle is greater.; 어떤 삼각형이든 마주보는 각이 더 큰 변이 더 크다.
Proposition 20.: In any triangle the sum of any two sides is greater than the remaining one.; 어떤 삼각형이든 두 변의 합은 나머지 한 변보다 크다.
Proposition 21.: If from the ends of one of the sides of a triangle two straight lines are constructed meeting within the triangle, then the sum of the straight lines so constructed is less than the sum of the remaining two sides of the triangle, but the constructed straight lines contain a greater angle than the angle contained by the remaining two sides.; 한 변의 양 끝 점에서 삼각형 안에서 만나는 두 직선을 그으면, 두 직선이 만든 변의 합은 다른 두 변의 합보다 크고 두 직선이 이루는 각은 두 변이 이루는 각보다 크다.
Proposition 22.: To construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one.; 주어진 세 선분으로 이루어진 삼각형을 그릴 수 있다. 단, 두 선분의 합이 나머지 다른 변보다 커야만 한다.
Proposition 23.: To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it.; 한 직선 위의 점에서 주어진 각과 같은 각을 그릴 수 있다.
Proposition 24.: If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base.; 두 변이 서로 같은 두 삼각형에서 두 변이 이루는 각이 어느 한쪽이 크다면 마찬가지로 밑면도 한 쪽이 크다.
Proposition 25.: If two triangles have two sides equal to two sides respectively, but have the base greater than the base, then they also have the one of the angles contained by the equal straight lines greater than the other.; 두 변이 서로 같은 두 삼각형에서 밑변은 어느 한쪽이 더 크다면 두 변이 이루는 각도 어느 한 쪽이 더 크다.
Proposition 26.: If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle.; 두 각이 각각 같고 한 변이 같은 두 삼각형은(서로 같은 변이 두 각과 모두 만나거나 건너편에 있는 변이라도) 나머지 각과 변도 서로 같다.
Proposition 27.: If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.; 두 직선이 다른 한 직선과 만나서 이루는 엇각이 서로 같다면, 두 직선은 서로 평행하다.
Proposition 28.: If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another.; 두 직선과 다른 한 직선이 만나서 이루는 동위각이 서로 같거나 같은 쪽에 있는 내각의 합이 180이면, 두 직선은 서로 평행하다.
Proposition 29.: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.; 평행한 두 직선과 한 직선이 만나서 이루는 엇각과 동위각은 서로 같고 같은 쪽 내각의 합은 180이다.
Proposition 30.: Straight lines parallel to the same straight line are also parallel to one another.; 같은 직선에 평행한 두 직선은 서로 평행하다.
Proposition 31.: To draw a straight line through a given point parallel to a given straight line.; 주어진 점을 지나고 주어진 직선과 평행인 직선을 그을 수 있다.
Proposition 32.: In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.; 어떤 삼각형이든 한 변으로 만들어지는 외각은 맞은 편 두 내각의 합과 같고 삼각형 세 내각의 합은 180이다.
Proposition 33.: Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.; 평행하고 길이가 같은 두 선분의 끝 점을 같은 방향으로 이은 두 직선은 서로 평행하다.
Proposition 34.: In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.; 평행사변형에서 마주 보는 변과 각은 서로 같고 대각선은 넓이를 이등분한다.
Proposition 35.: Parallelograms which are on the same base and in the same parallels equal one another.; 같은 밑변을 가지고 같은 평행선을 가지는 평행사변형은 넓이가 같다.
Proposition 36.: Parallelograms which are on equal bases and in the same parallels equal one another.; 밑변의 길이가 같고 같은 평행선을 가지는 평행사변형은 넓이가 같다.
Proposition 37.: Triangles which are on the same base and in the same parallels equal one another.; 같은 밑변을 가지고 꼭짓점이 같은 평행선 위에 있는 두 삼각형은 넓이가 같다.
Proposition 38.: Triangles which are on equal bases and in the same parallels equal one another.; 밑변 길이가 같고 꼭짓점이 같은 평행선 위에 있는 두 삼각형은 넓이가 같다.
Proposition 39.: Equal triangles which are on the same base and on the same side are also in the same parallels.; 공통 밑변을 가지고 같은 쪽에 꼭짓점이 있는 넓이가 같은 두 삼각형은 같은 평행선 위에 꼭짓점이 있다.
Proposition 40.: Equal triangles which are on equal bases and on the same side are also in the same parallels.; 밑변 길이가 같고 같은 쪽에 꼭짓점이 있는 넓이가 같은 삼각형은 서로 평행하다.
Proposition 41.: If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.; 평행사변형이 삼각형과 같은 밑변을 가지고 평행이면 평행사변형은 삼각형 크기의 두 배이다.
Proposition 42.: To construct a parallelogram equal to a given triangle in a given rectilinear angle.; 주어진 각을 가지는 삼각형과 넓이가 같고 같은 각을 가진 평행사변형을 만들 수 있다.
Proposition 43.: In any parallelogram the complements of the parallelograms about the diameter equal one another.; 평행사변형 안에 대각선 길이가 같은 평행사변현 쌍이 있다.
Proposition 44.: To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.; 어떤 직선, 각 그리고 삼각형을 주었을 때, 주어진 직선과 각을 가지며 삼각형과 넓이가 같은 평행사변현을 만들어라.
Proposition 45.: To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.; 어떤 다각형과 각을 주었을 때, 그 다각형과 넓이가 같고 주어진 각을 가지는 평행사변형을 만들어라.
Proposition 46.: To describe a square on a given straight line.; 주어진 선분을 한 변으로 하는 정사각형을 만들어라.
Proposition 47.: In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.; 직각삼각형에서 직각과 마주 보는 변을 가지고 정사각형을 만들면, 넓이는 나머지 다른 변으로 만든 정사각형 넓이를 더한 것과 같다.
Proposition 48.: If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.; 어떤 삼각형이 있는데, 한 변을 가지고 만든 정사각형의 넓이가 다른 변으로 만든 두 정사각형 넓이를 더한 것과 같다면 다른 두 변 사이의 각은 직각이다.

About the Definitions

The Elements begins with a list of definitions. Some of these indicate little more than certain concepts will be discussed, such as Def.I.1, Def.I.2, and Def.I.5, which introduce the terms point, line, and surface. (Note that for Euclid, the concept of line includes curved lines.) Others are substantial definitions which actually describe new concepts in terms of old ones. For example, Def.I.10 defines a right angle as one of two equal adjacent angles made when one straight line meets another. Other definitions look like they’re substantial, but actually are not. For instance, Def.I.4 says a straight line “is a line which lies evenly with the points on itself.” No where in the Elements is the defining phrase “which lies evenly with the points on itself” applicable. Thus, this definition indicates, at most, that some lines under discussion will be straight lines.

It has been suggested that the definitions were added to the Elements sometime after Euclid wrote them. Another possibility is that they are actually from a different work, perhaps older. In Def.I.22 special kinds of quadrilaterals are defined including square, oblong (a rectangle that are not squares), rhombus (equilateral but not a square), and rhomboid (parallelogram but not a rhombus). Except for squares, these other shapes are not mentioned in the Elements. Euclid does use parallelograms, but they’re not defined in this definition. Also, the exclusive nature of some of these terms—the part that indicates not a square—is contrary to Euclid’s practice of accepting squares and rectangles as kinds of parallelograms.

정의에 대하여

원론은 정의들로 시작한다. 점, 선, 면을 소개하는 정의(Def.I.1, Def.I.2, and Def.I.5)와 같은 것들은 논의해야 할 더 분명한 개념을 가리킨다.(유클리드가 말하는 선은 곡선을 포함하고 있다.) 다른 것들은 앞에 쓰인 개념으로 새로운 개념을 실제로 기술하는 본질적인 정의이다. 예를 들면 정의(Def.I.10)는 직각을 만나는 두 직선이 있을 때 이웃하는 두 각이 같은 것으로 정의하였다. 다른 것들은 본질적으로 보이지만 그렇지 않다. 예를 들면 정의(Def.I.4)는 직선을 점이 고르게 놓여 있는 것으로 말하지만 책 어디에도 점이 고르게 놓인 것을 적절하게 기술하여 직선을 정의할 문구가 없다. 그래서 이 정의는 기껏해야 검토하는 몇몇의 선이 직선이 됨을 가리킨다.

유클리드가 원론을 쓰고 난 다음에 덧붙여진 정의가 있다고 제안되었다. 더 오래된 다른 책으로부터 받아들여졌을 가능성이 있다. 정의(Def.I.22)는 정사각형, 직사각형, 마름모, 마름모가 아닌 평행사변형과 같은 특별한 사변형을 정의한다. 유클리드는 평행사변형(parallelogram)이란 용어를 사용하지만 이 정의에서 따로 정의하지 않았다. 또한 직사각형을 말하며 -정사각형이 아니라는- 배타적인 성질은 정사각형과 직사각형을 평행사변형으로 받아들이는 유클리드의 연구와 배치된다.(정사각형은 직사각형에 포함된다.)

About the Postulates

Following the list of definitions is a list of postulates. Each postulate is an axiom—which means a statement which is accepted without proof— specific to the subject matter, in this case, plane geometry. Most of them are constructions. For instance, Post.I.1 says a straight line can be drawn between two points, and Post.I.3 says a circle can be drawn given a specified point to be the center and another point to be on the circumference. The fourth postulate, Post.I.4, is not a constuction, but says that all right angles are equal.

공준에 대하여

정의 다음에 공준이 있다. 각각의 공준은 평면기하를 구성하는 공리(증명없이 참으로 받아들이는 명제)이다. 공준들은 모두 구조가 있다. 공준(Post.I.1)는 두 점 사이를 잇는 직선을 그릴 수 있다고 말한다. 공준(Post.I.3)는 주어진 한 점을 중심으로 하고 다른 점은 원주에 있는 원을 그릴 수 있음을 말한다. 네 번째 공준(Post.I.4)는 구조가 없고 모든 직각은 같다고 말한다.

About magnitudes and the Common Notions

The Common Notions are also axioms, but they refer to magnitudes of various kinds. The kind of magnitude that appears most frequently is that of straight line. Other important kinds are rectilinear angles and areas (plane figures). Later books include other kinds.

In proposition III.16 (but nowhere else) angles with curved sides are compared with rectilinear angles which shows that rectilinear angles are to be considered as a special kind of plane angle. That agrees with Euclid’s definition of them in I.Def.9 and I.Def.8.

Also in Book III, parts of circumferences of circles, that is, arcs, appear as magnitudes. Only arcs of equal circles can be compared or added, so arcs of equal circles comprise a kind of magnitude, while arcs of unequal circles are magnitudes of different kinds. These kinds are all different from straight lines. Whereas areas of figures are comparable, different kinds of curves are not.

Book V includes the general theory of ratios. No particular kind of magnitude is specified in that book. It may come as a surprise that ratios do not themselves form a kind of magnitude since they can be compared, but they cannot be added. See the guide on Book V for more information.

Number theory is treated in Books VII through IX. It could be considered that numbers form a kind of magnitude as pointed out by Aristotle.

Beginning in Book XI, solids are considered, and they form the last kind of magnitude discussed in the Elements.

크기와 공통관념에 대하여

공통관념도 공준과 마찬가지로 공리지만 다양한 꼴로 크기를 말하고 있다. 가장 자주 나타나는 크기는 선분의 길이다. 다른 중요한 것은 직선각의 크기와 평면 도형의 넓이다. 뒤에 있는 책은 또 다른 크기를 포함하고 있다.

명제(III.16)에는 곡면 사이의 각을 평면 사이의 직선각을 보이는 것처럼 직선각과 비굫고 있다. 이것은 정의 ( I.Def.9 and I.Def.8. )와 일치한다.

또한 3권에서 원주의 일부, 즉 호의 길이가 나온다. 반지름이 같은 원에 있는 호의 길이는 서로 비교하거나 더할 수 있으나 반지름이 다른 원에 있는 호는 그럴 수 없다. 호의 길이와 같은 크기는 직선의 길이와는 다르다. 도형의 넓이는 비교 가능하지만 곡선의 길이는 가능하지 않다.

5권은 비율의 일반 정리를 포함하고 있다. 지정된 특별한 크기는 없다. 비율은 서로 비교할 수 있지만 더 할 수는 없기에 스스로 크기의 한 종류가 되지 못하는 것은 놀라운 일이다.

정수론은 7권에서 9권까지 다루어진다. 아리스토텔레스가 강조한 크기의 한 종류에서 나오는 수를 고려하고 있다.

11권은 원론에서 다루는 마지막 크기인 입체의 부피를 다루는 걸로 시작한다.

The propositions

Following the definitions, postulates, and common notions, there are 48 propositions. Each of these propositions includes a statement followed by a proof of the statement. Each statement of the proof is logically justified by a definition, postulate, common notion, or an earlier proposition that has already been proven. There are gaps in the logic of some of the proofs, and these are mentioned in the commenaries after the propositions. Also included in the proof is a diagram illustrating the proof.

Some of the propositions are constructions. A construction depends, ultimately, on the constructive postulates about drawing lines and circles. The first part of a proof for a constuctive proposition is how to perform the construction. The rest of the proof (usually the longer part), shows that the proposed construction actually satisfies the goal of the proposition. In the list of propositions in each book, the constructions are displayed in red.

Most of the propositions, however, are not constructions. Their statements say that under certain conditions, certain other conditions logically follow. For example, Prop.I.5 says that if a triangle has the property that two of its sides are equal, then it follows that the angles opposite these sides (called the “base angles”) are also equal. Even the propositions that are not constructions may have constructions included in their proofs since auxillary lines or circles may be needed in the explanation. But the bulk of the proof is, as for the constructive propositions, a sequence of statements that are logically justified and which culminates in the statement of the proposition.

명제

정의, 공준, 공통관념에 이어 명제 48가지가 나온다. 이 명제들은 증명 과정을 포함하고 있다. 모든 증명 과정은 정의, 공준, 공통관념 그리고 앞에서 이미 증명된 명제에 의해 논리적으로 보여진다. 논리에 빈틈이 있는 어떤 증명은 뒤에 나오는 명제에서 주석을 달고 있다. 또한 증명을 보여주는 그림을 포함하고 있다.

몇몇 명제는 작도이다. 작도는 궁극적으로 직선과 원을 작도하는 공준에 달려 있다. 작도하는 명제를 증명하는 첫 부분은 어떻게 작도 하는가를 나타낸다. 나머지 부분은 작도가 명제가 목표로 한 것을 제대로 만족하는 가를 확인하고 있다. 각 권에 있는 명제 목록에는 작도가 붉은 색으로 보여지고 있다.

명제 대부분은 작도가 아니다. 그들은 확실한 조건, 논리적으로 따르는 확실한 조건을 말한다. 예를 들면 명제(Prop.I.5)는 ‘이등변 삼각형은 밑각도 서로 같다.’이다. 작도하는 명제가 아니더라도 설명 속에 실제로는 직선과 원을 작도하는 것을 포함하고 있다. 대부분의 증명은 작도하는 명제로 주어진 명제가 주장하는 바가 참임을 보이는 과정에 결정적인 순간이 있다.

Logical structure of Book I

The various postulates and common notions are frequently used in Book I. Only two of the propositions rely solely on the postulates and axioms, namely, I.1 and I.4. The logical chains of propositions in Book I are longer than in the other books; there are long sequences of propositions each relying on the previous.

1권의 논리적 구조

1권에는 다양한 공준과 공통관념이 자주 사용된다. 딱 두 가지 명제만이 공준과 공리(I.1 and I.4)에 전적으로 의존한다. 1권에 있는 명제의 논리 사슬은 다른 권보다 훨씬 길다. 앞선 명제에 의존하는 긴 과정이 있다.

작성자: veritas

7권

Table of contents

Definitions

Propositions

6권

Table of contents

Definitions

Propositions

Logical structure of Book VI

5권

Propositions

Guide for Book V

Background on ratio and proportion

Summary of the propositions

Logical structure of Book V

4권

Propositions

Guide to Book IV

Logical structure of Book IV

3권

Propositions

2권

Propositions

Guide to Book II

Logical structure of Book II

1권

Table of contents

Postulates 공준

Common Notions 공통 관념

Propositions 명제

About the Definitions

About the Postulates

About magnitudes and the Common Notions

The propositions

Logical structure of Book I