Euclid's Elements

OF THOSE THINGS WHICH ARE

demonstrated in these books besides those that are Euclid's.

IN THE FIRST BOOK.

A decorative rectangular woodcut serves as the initial letter 'C'. It depicts a bearded scholar in academic robes and a hat, seated at a desk. He is using a pair of dividers on a globe. Books and scientific instruments are visible in the background.

The diameter of a circle bisects the circle. 3.b
To construct an isosceles and scalene triangle on a given straight line. 8
If two straight lines not taken on the same sides make vertical angles equal to two straight lines, those straight lines will be in a straight line with each other. 14.b
If a certain straight line cuts one of the parallels, it will also cut the other. 19.b
A straight line, which is produced to infinity from less than two right angles, meets the other. 20
Every rectilinear figure has the angles constructed outside equal to four right angles. 21
Every quadrilateral which has opposite sides and opposite angles equal is a parallelogram. 22
Every quadrilateral which is bisected by both diameters is a parallelogram. 24
If a triangle is double a parallelogram and they have the same base or equal bases, and they are on the same sides, they will also be in the same parallels. 25
If a triangle is double a parallelogram and both are in the same parallels, they will be either on one and the same base or on equal ones. 25
How a parallelogram equal to a given rectilinear figure can be applied to a given straight line in a given rectilinear angle. 26.b
Squares described from equal straight lines are also equal to each other. 27
Squares equal to each other are described from equal straight lines.
From two straight lines, which are equal to two given lines, to construct a parallelogram in a given rectilinear angle. 27.b

IN THE SECOND BOOK.

If there are two straight lines which are cut into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by each part of one applied to each part of the other. 29.b
If there are two straight lines which are cut in any way, the rectangle contained by the wholes together with that which is contained by two of their parts is equal to the rectangles which are contained by the wholes and the said parts together with that which is contained by the remaining parts.
Arithmetical analogue proportional demonstration. However, it is a theorem.
The square which is made from the excess together with that which is contained by the extremes is equal to the square of the mean. 31.b
If a straight line is cut into unequal parts, the squares of those parts are equal to the rectangle which is contained twice by the said parts, together with the square of the line by which the greater part exceeds the smaller. 32
Proposition IX is demonstrated otherwise. 33
Proposition X is demonstrated otherwise. 33.b
To measure the area of any triangle having an obtuse angle. 34
The converse of Proposition XIII. 35
To find the area of any triangle, whether acute-angled, right-angled, or obtuse-angled, that has known sides. 35.b

IN THE THIRD BOOK.

The converse of the definition of a circle: if equal straight lines fall from any point of those which are inside to the boundary of a figure, that is a circle. 37.b. 38
The converse of Proposition VII. 39.b
If any point is taken on the circumference of a circle, and straight lines are drawn from it into the circle, that which passes through the center will be the greatest of all; of the others, those that are nearer to the one passing through the center are greater than those farther away, and only two are equal on both sides of the greatest. 40
The converse of Proposition XIX. 43.b