"As for ancient geometrical analysis and modern algebra, even apart from the fact that they deal only in highly abstract matters that seem to have no practical application, the former is so closely tied to the consideration of figures that it is unable to exercise the intellect without greatly tiring the imagination, while in the latter case one is so much a slave to certain rules and symbols that it has been turned into a confused and obscure art that bewilders the mind instead of being a form of knowledge that cultivates it."
— R. Descartes, A Discourse on the Method, 1637
"... geometry is nothing other than that part of universal mechanics which reduces the art of measuring to exact propositions and demonstrations. But geometry is commonly used in reference to magnitude, and mechanics in reference to motion. In this sense rational mechanics will be the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever."
— I. Newton, Preface to Principia, 1687
Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Interesting construction: Solid angle
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
Based on this translation.
Care by ratherthanpaper as code to be read.
1. A point is that of which there is no part... continue
1. Let it have been postulated to draw a straight-line from any point to any point... continue
1. Things equal to the same thing are also equal to one another... continue
To construct an equilateral triangle on a given finite straight-line... continue
To place a straight-line equal to a given straight-line at a given point... continue
For two given unequal straight-lines, to cut off from the greater a straight-line equal to the lesser... continue
If two triangles have two corresponding sides equal, and have the angles enclosed by the equal sides equal, then they will also have equal bases, and the two triangles will be equal, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles... continue
For isosceles triangles, the angles at the base are equal to one another, and if the equal sides are produced then the angles under the base will be equal to one another... continue
If a triangle has two angles equal to one another then the sides subtending the equal angles will also be equal to one another... continue
On the same straight-line, two other straight-lines equal, respectively, to two (given) straight-lines (which meet) cannot be constructed (meeting) at a different point on the same side (of the straight-line), but having the same ends as the given straight-lines... continue
If two triangles have two corresponding sides equal, and also have equal bases, then the angles encompassed by the equal straight-lines will also be equal... continue
To cut a given rectilinear angle in half... continue
To cut a given finite straight-line in half... continue
To draw a straight-line at right-angles to a given straight-line from a given point on it... continue
To draw a straight-line perpendicular to a given infinite straight-line from a given point which is not on it... continue
If a straight-line stood on a(nother) straight-line makes angles, it will certainly either make two right-angles, or (angles whose sum is) equal to two right-angles... continue
If two straight-lines, not lying on the same side, make adjacent angles (whose sum is) equal to two right-angles at the same point on some straight-line, then the two straight-lines will be straight-on (with respect) to one another... continue
If two straight-lines cut one another then they make the vertically opposite angles equal to one another... continue
For any triangle, when one of the sides is produced, the external angle is greater than each of the internal and opposite angles... continue
For any triangle, (the sum of any) two angles is less than two right-angles, (the angles) being taken up in any (possible way)... continue
For any triangle, the greater side subtends the greater angle... continue
For any triangle, the greater angle is subtended by the greater side... continue
For any triangle, (the sum of any) two sides is greater than the remaining (side), (the sides) being taken up in any (possible way)... continue
If two internal straight-lines are constructed on one of the sides of a triangle, from its ends, the constructed (straight-lines) will be less than the two remaining sides of the triangle, but will encompass a greater angle... continue
To construct a triangle from three straight-lines which are equal to three given [straight-lines]... continue
To construct a rectilinear angle equal to a given rectilinear angle at a (given) point on a given straight-line... continue
If two triangles have two sides equal to two sides, respectively, but (one) has the angle encompassed by the equal straight-lines greater than the (corresponding) angle (in the other), then (the former triangle) will also have a base greater than the base (of the latter)... continue
If two triangles have two sides equal to two sides, respectively, but (one) has a base greater than the base (of the other), then (the former triangle) will also have the angle encompassed by the equal straight-lines greater than the (corresponding) angle (in the latter)... continue
If two triangles have two angles equal to two angles, respectively, and one side equal to one side ---in fact, either that by the equal angles, or that subtending one of the equal angles--- then (the triangles) will also have the remaining sides equal to the [corresponding] remaining sides, and the remaining angle (equal) to the remaining angle... continue
If a straight-line falling across two straight-lines makes the alternate angles equal to one another then the (two) straight-lines will be parallel to one another... continue
If a straight-line falling across two straight-lines makes the external angle equal to the internal and opposite angle on the same side, or (makes) the (sum of the) internal (angles) on the same side equal to two right-angles, then the (two) straight-lines will be parallel to one another... continue
A straight-line falling across parallel straight-lines makes the alternate angles equal to one another, the external (angle) equal to the internal and opposite (angle), and the (sum of the) internal (angles) on the same side equal to two right-angles... continue
(Straight-lines) parallel to the same straight-line are also parallel to one another... continue
To draw a straight-line parallel to a given straight-line, through a given point... continue
For any triangle, (if) one of the sides (is) produced (then) the external angle is equal to the (sum of the) two internal and opposite (angles), and the (sum of the) three internal angles of the triangle is equal to two right-angles... continue
Straight-lines joining equal and parallel (straight-lines) on the same sides are themselves also equal and parallel... continue
For parallelogrammic figures, the opposite sides and angles are equal to one another, and a diagonal cuts them in half... continue
Parallelograms which are on the same base and between the same parallels are equal to one another... continue
Parallelograms which are on equal bases and between the same parallels are equal to one another... continue
Triangles which are on the same base and between the same parallels are equal to one another... continue
Triangles which are on equal bases and between the same parallels are equal to one another... continue
Equal triangles which are on the same base, and on the same side, are also between the same parallels... continue
Equal triangles which are on equal bases, and on the same side, are also between the same parallels... continue
If a parallelogram has the same base as a triangle, and is between the same parallels, then the parallelogram is double (the area) of the triangle... continue
To construct a parallelogram equal to a given triangle in a given rectilinear angle... continue
For any parallelogram, the complements of the parallelograms about the diagonal are equal to one another... continue
To apply a parallelogram equal to a given triangle to a given straight-line in a given rectilinear angle... continue
To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle... continue
To describe a square on a given straight-line... continue
In a right-angled triangle, the square on the side subtending the right-angle is equal to the (sum of the) squares on the sides surrounding the right-angle... continue
If the square on one of the sides of a triangle is equal to the (sum of the) squares on the remaining sides of the triangle then the angle contained by the remaining sides of the triangle is a right-angle... continue
Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
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