hcju/proofwikips huggingface dataset:
Definition:Euclid's Definitions - Book I/12 - Acute Angle
An '''acute angle''' is an angle less than a right angle.
Definition:Euclid's Definitions - Book I/13 - Boundary
A '''boundary''' is that which is an extremity of anything.
Definition:Euclid's Definitions - Book I/14 - Geometric Figure
A '''figure''' is that which is contained by any boundary or boundaries.
Definition:Euclid's Definitions - Book I/15 - Circle
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 are equal to one another;
Definition:Euclid's Definitions - Book I/16 - Center of Circle
And the point is called the '''center of the circle'''.
Definition:Euclid's Definitions - Book I/17 - Diameter of Circle
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 center.
Definition:Euclid's Definitions - Book I/18 - Semicircle
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.
Definition:Euclid's Definitions - Book I/19 - Rectilineal Figure
'''Rectilineal figures''' are those which are contained by straight lines, '''trilateral''' figures being those contained by three, '''quadrilateral''' those contained by four, and '''multi-lateral''' those contained by more than four straight lines.
Definition:Euclid's Definitions - Book I/22 - Quadrilaterals
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 equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called '''trapezia'''.
Definition:Euclid's Definitions - Book I/20 - Triangles by Sides
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.
Definition:Euclid's Definitions - Book I/21 - Triangles by Angles
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.
Definition:Euclid's Definitions - Book I/23 - Parallel Lines
'''Parallel''' straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.
Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle
Any rectangular parallelogram is said to be '''contained''' by the two straight lines containing the right angle.
Definition:Euclid's Definitions - Book II/2 - Gnomon
And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a '''gnomon'''.
Definition:Euclid's Definitions - Book III/1 - Equal Circles
'''Equal circles''' are those the diameters of which are equal, or the radii of which are equal.
Definition:Euclid's Definitions - Book III/2 - Tangent to Circle
A straight line is said to '''touch a circle''' which, meeting the circle and being produced, does not cut the circle.
Definition:Euclid's Definitions - Book III/3 - Tangent Circles
Circles are said to '''touch one another''' which, meeting one another, do not cut one another.
Definition:Euclid's Definitions - Book III/4 - Equally Distant in Circle
In a circle straight lines are said to be '''equally distant from the center''' when the perpendiculars drawn to them from the center are equal.
Definition:Euclid's Definitions - Book III/5 - Greater Distance in Circle
And that straight line is said to be '''at a greater distance''' on which the greater perpendicular falls.
Definition:Euclid's Definitions - Book III/6 - Segment of Circle
A '''segment of a circle''' is the figure contained by a straight line and a circumference of a circle.
Definition:Euclid's Definitions - Book III/7 - Angle of Segment
An '''angle of a segment''' is that contained by a straight line and a circumference of a circle.
Definition:Euclid's Definitions - Book III/8 - Angle in Segment
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 extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.
Definition:Euclid's Definitions - Book III/9 - Stand on Circumference
And, when the straight lines containing the angle cut off a circumference, the angle is said to '''stand upon''' that circumference.
Definition:Euclid's Definitions - Book III/10 - Sector of Circle
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.
Definition:Euclid's Definitions - Book III/11 - Similar Segments
'''Similar segments of circles''' are those which admit equal angles, or in which the angles are equal to one another.
Definition:Euclid's Definitions - Book IV/1 - Inscribe
A rectilineal figure is said to be '''inscribed in a rectilineal figure''' when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.
Definition:Euclid's Definitions - Book IV/2 - Circumscribe
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.
Definition:Euclid's Definitions - Book IV/3 - Inscribed in Circle
A rectilineal figure is said to be '''inscribed in a circle''' when each angle of the inscribed figure lies on the circumference of the circle.
Definition:Euclid's Definitions - Book IV/4 - Circumscribed about Circle
A rectilineal figure is said to be '''circumscribed about a circle''', when each side of the circumscribed figure touches the circumference of the circle.
Definition:Euclid's Definitions - Book IV/5 - Circle Inscribed
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.
Definition:Euclid's Definitions - Book IV/6 - Circle Circumscribed
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.
Definition:Euclid's Definitions - Book IV/7 - Fitted Into Circle
A straight line is said to be '''fitted into a circle''' when its extremities are on the circumference of the circle.
Definition:Measure (Geometry)
A geometric quantity $A$ is said to '''measure''' another quantity $B$ when the size of $A$ is a divisor of the size of $B$.
Definition:Part
=== Of Integer === {{:Definition:Divisor (Algebra)/Integer/Aliquot Part}} === Of Real Number === {{:Definition:Divisor (Algebra)/Real Number/Part}}
Definition:Euclid's Definitions - Book V/1 - Part
A magnitude is a '''part''' of a magnitude, the less of the greater, when it measures the greater.
Definition:Euclid's Definitions - Book V/2 - Multiple
The greater is a '''multiple''' of the less when it is measured by the less.
Definition:Euclid's Definitions - Book V/3 - Ratio
A '''ratio''' is a sort of relation in respect of size between two magnitudes of the same kind.
Definition:Euclid's Definitions - Book V/4 - Existence of Ratio
Magnitudes are said to '''have a ratio''' to one another which are capable, when multiplied, of exceeding one another.
Definition:Euclid's Definitions - Book V/5 - Equality of Ratios
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 be 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.
Definition:Euclid's Definitions - Book V/6 - Proportion
Let magnitudes which have the same ratio be called '''proportional'''.
Definition:Euclid's Definitions - Book V/7 - Greater Ratio
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:Euclid's Definitions - Book V/8 - Proportion in Three Terms
A proportion in three terms is the least possible.
Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio
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:Euclid's Definitions - Book V/10 - Triplicate Ratio
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:Term (Algebra)
A '''term''' is either a variable or a constant. Let $a \circ b$ be an expression. Then each of $a$ and $b$ are known as the '''terms''' of the expression. The word '''term''' is usually used when the operation $\circ$ is addition, that is $+$.
Definition:Term (Natural Language)
A '''term''' is a noun which is assigned a specified definition within mathematics. In a more specialized context, the word '''term''' is used for an element of an expression.
Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes
The term '''corresponding magnitudes''' is used of antecedents in relation to antecedents, and of consequents in relation to consequents.
Definition:Euclid's Definitions - Book V/12 - Alternate Ratio
'''Alternate ratio''' means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
Definition:Euclid's Definitions - Book V/13 - Inverse Ratio
'''Inverse ratio''' means taking the consequent as antecedent in relation to the antecedent as consequent.
Definition:Euclid's Definitions - Book V/14 - Composition of Ratio
'''Composition of a ratio''' means taking the antecedent together with the consequent as one in relation to the consequent by itself.
Definition:Euclid's Definitions - Book V/15 - Separation of Ratio
'''Separation of a ratio''' means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali
A '''ratio ex aequali''' arises when, there being several magnitudes and another set equal to them in multitude which makes two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;<br/>Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio
'''Conversion of a ratio''' means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion
A '''perturbed proportion''' arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.
Definition:Euclid's Definitions - Book VI/1 - Similar Rectilineal Figures
'''Similar rectilineal figures''' are such as have their angles severally equal and the sides about the equal angles proportional.
Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures
Two figures are '''reciprocally related''' when there are in each of the two figures antecedent and consequent ratios.
Definition:Euclid's Definitions - Book VI/3 - Extreme and Mean Ratio
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:Euclid's Definitions - Book VI/4 - Height
The '''height''' of any figure is the perpendicular drawn from the vertex to the base.
Definition:Euclid's Definitions - Book VII/1 - Unit
An '''unit''' is that of which each of the things that exist is called one.
Definition:Euclid's Definitions - Book VII/2 - Number
A '''number''' is a multitude composed of units.
Definition:Euclid's Definitions - Book VII/3 - Part
A number is '''a part''' of a number, the less of the greater, when it measures the greater;
Definition:Euclid's Definitions - Book VII/4 - Parts
but '''parts''' when it does not measure it.
Definition:Euclid's Definitions - Book VII/5 - Multiple
The greater number is a '''multiple''' of the less when it is measured by the less.
Definition:Euclid's Definitions - Book VII/6 - Even Number
An '''even number''' is that which is divisible into two equal parts.
Definition:Euclid's Definitions - Book VII/7 - Odd Number
An '''odd number''' is that which is not divisible into two equal parts, or that which differs by an unit from an even number.
Definition:Euclid's Definitions - Book VII/8 - Even Times Even
An '''even-times even number''' is that which is measured by an even number according to an even number.
Definition:Euclid's Definitions - Book VII/9 - Even Times Odd
An '''even-times odd number''' is that which is measured by an even number according to an odd number.
Definition:Euclid's Definitions - Book VII/10 - Odd Times Odd
An '''odd-times odd number''' is that which is measured by an odd number according to an odd number.
Definition:Euclid's Definitions - Book VII/11 - Prime Number
A '''prime number''' is that which is measured by an unit alone.
Definition:Euclid's Definitions - Book VII/12 - Relatively Prime
Numbers '''prime to one another''' are those which are measured by an unit alone as a common measure.
Definition:Euclid's Definitions - Book VII/13 - Composite Number
A '''composite number''' is that which is measured by some number.
Definition:Euclid's Definitions - Book VII/14 - Relatively Composite
Numbers '''composite to one another''' are those which are measured by some number as a common measure.
Definition:Euclid's Definitions - Book VII/15 - Multiply
A number is said to '''multiply''' a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
Definition:Euclid's Definitions - Book VII/16 - Plane Number
And, when two numbers having multiplied one another make some number, the number so produced is called '''plane''', and its '''sides''' are the numbers which have multiplied one another.
Definition:Euclid's Definitions - Book VII/17 - Solid Number
And, when three numbers having multiplied one another make some number, the number so produced is '''solid''', and its '''sides''' are the numbers which have multiplied one another.
Definition:Euclid's Definitions - Book VII/18 - Square Number
A '''square number''' is equal multiplied by equal, or a number which is contained by two equal numbers.
Definition:Euclid's Definitions - Book VII/19 - Cube Number
And a '''cube''' is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
Definition:Euclid's Definitions - Book VII/20 - Proportional
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:Euclid's Definitions - Book VII/21 - Similar Numbers
'''Similar plane''' and '''solid''' numbers are those which have their sides proportional.
Definition:Euclid's Definitions - Book VII/22 - Perfect Number
A '''perfect number''' is that which is equal to its own parts.
Definition:Euclid's Definitions - Book X/1 - Commensurable
Those magnitudes are said to be '''commensurable''' which are measured by the same same measure, and those '''incommensurable''' which cannot have any common measure.
Definition:Euclid's Definitions - Book X/2 - Commensurable in Square
Straight lines are '''commensurable in square''' when the squares on them are measured by the same area, and '''incommensurable in square''' when the squares on them cannot possibly have any area as a common measure.
Definition:Euclid's Definitions - Book X/3 - Rational Line Segment
With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called '''rational''', and those straight lines which are commensurable with it, whether in length and in square or square only, '''rational''', but those which are incommensurable with it '''irrational'''.
Definition:Euclid's Definitions - Book X/4 - Rational Area
And let the square on the assigned straight line be called '''rational''' and those areas which are commensurable with it '''rational''', but those which are incommensurable with it '''irrational''', and the straight lines which produce them '''irrational''', that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.
Definition:Euclid's Definitions - Book X (II)/1 - First Binomial
Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''first binomial''' straight line;
Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial
but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a '''second binomial''';
Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial
and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a '''third binomial'''.
Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial
Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''fourth binomial''';
Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial
if the lesser, a '''fifth binomial''';
Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial
and if neither, a '''sixth binomial'''.
Definition:Euclid's Definitions - Book X (III)/1 - First Apotome
Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''first apotome'''.
Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome
But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable in length with the whole, let the apotome be called a '''second apotome'''.
Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome
But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a '''third apotome'''.
Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome
Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''fourth apotome''';
Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome
if the annex be so commensurable, a '''fifth''';
Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome
and if neither, a '''sixth'''.
Definition:Euclid's Definitions - Book XI/1 - Solid
A '''solid''' is that which has length, breadth, and depth.
Definition:Euclid's Definitions - Book XI/3 - Line at Right Angles to Plane
A '''straight line''' is '''at right angles to a plane''' when it makes right angles with all the straight lines which meet it and are in the plane.
Definition:Euclid's Definitions - Book XI/4 - Plane at Right Angles to Plane
A '''plane''' is '''at right angles to a plane''' when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.
Definition:Euclid's Definitions - Book XI/5 - Inclination of Straight Line
The '''inclination of a straight line to a plane''' is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.
Definition:Euclid's Definitions - Book XI/6 - Inclination of Plane
The '''inclination of a plane to a plane''' is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.
Definition:Euclid's Definitions - Book XI/7 - Similarly Inclined
A plane is said to be '''similarly inclined''' to a plane as another is to another when the said angles of the inclinations are equal to one another.
Definition:Euclid's Definitions - Book XI/8 - Parallel Planes
'''Parallel planes''' are those which do not meet.
Definition:Euclid's Definitions - Book XI/9 - Similar Solid Figures
'''Similar solid figures''' are those contained by similar planes equal in multitude.
Definition:Euclid's Definitions - Book XI/10 - Similar Equal Solid Figures
'''Equal and similar solid figures''' are those contained by similar planes equal in multitude and in magnitude.
Definition:Euclid's Definitions - Book XI/11 - Solid Angle
A '''solid angle''' is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.<br/>Otherwise: A '''solid angle''' is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.
Definition:Euclid's Definitions - Book XI/12 - Pyramid
A '''pyramid''' is a solid figure, contained by planes, which is constructed from one plane to one point.
Definition:Euclid's Definitions - Book XI/13 - Prism
A '''prism''' is a solid figure contained by planes of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.
Definition:Euclid's Definitions - Book XI/14 - Sphere
When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''sphere'''.
Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere
The '''axis of the sphere''' is the straight line which remains fixed about which the semicircle is turned.
Definition:Euclid's Definitions - Book XI/16 - Center of Sphere
The '''center of the sphere''' is the same as that of the semicircle.
Definition:Euclid's Definitions - Book XI/17 - Diameter of Sphere
A '''diameter of the sphere''' is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.