## The Elements of Euclid; viz. the first six books,together with the eleventh and twelfth, with an appendix |

### From inside the book

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**less**" than two right angles , these straight lines being " produced , shall at length meet upon that side on " which are the angles which are**less**than two " right angles . " tulate . PROPOSITION I. PROBLEM . To describe an equilateral ... Page 9

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**less**. Let AB and C be the two given straight lines , whereof AB is the greater . It is required to cut off from AB , the greater , a part equal to C , the**less**. T E B * 2. 1 . From the point A draw✶ the straight line AD equal to C ... Page 21

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**less**than two right angles . Let ABC be any triangle ; any two of its angles to- gether shall be**less**than two right angles . Produce BC to D ; and be- cause ACD is BOOK I. PROP . XVII . 21. Page 22

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**less**than two right angles . In like manner , may be demonstrated , that BAC , ACB , as also CAB , ABC , are**less**than two right angles . Therefore any two angles , & c . Q. E. D. PROP . XVIII . THEOR . 3. 1 . * 5. 1 . The greater side ... Page 23

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**less**than it ; it is not equal , because then the angle ABC would be equal to the angle ACB ; but it * 5. 1 . is not ; therefore AC is not equal to AB : neither is it**less**; because then the angle ABC would be**less**than the angle ACB ...### Other editions - View all

The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ... Euclid No preview available - 2015 |

The Elements of Euclid: Viz. the First Six Books, Together With the Eleventh ... Euclid No preview available - 2023 |

### Common terms and phrases

AB is equal AC is equal altitude angle ABC angle ACB angle BAC base BC bisect centre circle ABCD circle EFGH circumference common section cone cylinder demonstrated described diameter draw equal to F equiangular equilateral equimultiples exterior angle fore given rectilineal given straight line gnomon inscribed join less Let ABC meet multiple opposite angle parallel parallelogram parallelopiped perpendicular polygon prisms PROB proved pyramid ABCG pyramid DEFH Q. E. D. PROP rectangle contained rectilineal figure remaining angle right angles segment solid angle solid CD sphere square of AC straight line AC THEOR third three plane angles three straight lines tiples touches the circle triangle ABC triangle DEF triplicate ratio twice the rectangle wherefore whole

### Popular passages

Page 173 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 56 - Iff a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 53 - If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

Page 58 - IF a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.

Page 94 - The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle.

Page 23 - Any two sides of a triangle are together greater than the third side.

Page 40 - EQUAL triangles upon the same base, and upon the same side of it, are between the same parallels.

Page 103 - If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it.

Page 50 - PROP. I. THEOR. If there oe two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines, is equal to the rectangles contained by the undivided line, and the several parts of the divided line.

Page 28 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.