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The paper reviews and connects old results of Archimedes, Eudoxus, Proclus, Aristotle, and Hilbert, and introduces Marvin's own result about Aristotle's Axiom. He truly was an extraordinary mathematical expositor, and had a deep interest in mathematical logic and the foundations of mathematics.

He was a quiet but jovial man whose heart was as big as the sky and had an appetite for life that matched. He had not only studied under the legendary Alan Watts, and noted that he dated his daughter. He spent two years living at a Buddhist retreat to study Vipassana. He had traveled the world. He had loved many women. Years ago, when we were new friends, he told me that he had met one of my kung fu masters for dinner in Chinatown. After he returned from the trip, he confided in me that it was actually a challenge for him to do things like travel. He was fearful. He was afraid to travel and afraid of change.

He was afraid to love and be loved fully. Most of all, he was afraid of depression. The unspoken reality about Marvin is that he had bravely and successfully fought depression most of his life. He had a truly powerful mind, but when it sank into depressive rumination, that power worked against him. He fought through these negative thoughts barehanded, and to some measure defeated despair and hopelessness. As a result, he ended up traveling the world and recovering from the heartbreak of a failed marriage to find a modicum of love.

Marvin derived so much meaning from his love life, that a recounting of the man would be woefully incomplete if it excluded his multilayered and passionate loves. Unlike other mathematicians, he was not impaired by shyness. I know a couple of mathematicians who are terminally shy and lifelong virgins. He told me later that sharing his darkest provocative secret with a woman liberated his heart, it made him feel worthy of love.

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He was very very fond of this woman, and deeply enjoyed being with a person who could understand his work. Another was a therapist and intellectual equal. In fact, so much of his attention and focus was spent on understanding love and passion that I once mentioned he should write a novel, rather than the thousands of pages of handwritten letters he sent to me with his many obsessions and confessions. I found his ramblings about life and love beautiful.

However, trying to truly understand love is a bit like trying to solve the Riemann hypothesis, profoundly difficult if not impossible. But still he tried, with each new love interest a prime number to explore lovingly. Some of his dalliances were only flirtations, but some were clearly dangerous. One woman from Vietnam, he admitted, was a disaster. However, most of his quiet affairs were simple flirtations and infatuations, contained and controlled, that gave his life meaning and helped him to fight off his depression.

He was like anyone, simply wishing for a little love and comfort in his declining years. The greatest love of his final years was a young Chinese nursing student he befriended. He tutored her earnestly and diligently, and she flourished under his tutelage. Tell us if something is incorrect.

Add to Cart. Free delivery. Arrives by Thursday, Oct Pickup not available. About This Item We aim to show you accurate product information. Manufacturers, suppliers and others provide what you see here, and we have not verified it. See our disclaimer. Customer Reviews. Write a review. See any care plans, options and policies that may be associated with this product. Email address. That part is lesser which has a greater denomination; and that part is greater which has a lesser denomination. The unity is a part of any number, and denominated by it. Every number is a multiple of the unity as many times as the unity is part of it.

Every number multiplied by the unity results in itself. The unity multiplied by any number results in this number. The difference of the extremes is composed of the differences of the same extremes to the mean term. To take any quantity of numbers equal to, or multiples of, any given number.

Campano da Novara, thirteenth century [cf. No number can be diminished to infinity. If a number is greater than another, and their difference is added to the lesser, or subtracted from the greater, the resulting numbers will be equal. Two numbers that have the same ratio to the same number are equal. A number may be subtracted from another number as many times as the former divides the latter. The greater number does not divide the lesser. If a given number is multiplied by another number and then is divided by it, the result is again the given number.

If a given number is divided by another number and then is multiplied by it, the result is again the given number. If a number measures a part [i. Candale, The sum of a given number with a number that is greater than another one is greater than its sum with the lesser number. Numbers which are made by the same quantity of unities are equal. Two number are equal if their parts with equal denomination are in equal quantity.

If two number are multiplied, the product is to the first as the second is to the unity. If a number divides another number, the divided number is to the divisor as the quotient is to the unity. If two numbers are added together which are greater than two other numbers, the sum of the greater ones is greater than the sum of the lesser ones. The ratio of a number with another number is the same as the ratio of the parts of the two numbers with the same denomination.

If a unity is added to an odd number, the result is an even number. If a unity is added to an even number, the result is an odd number. If several ratios among numbers are all equal to the same ratio, they will also be equal to one another. Billingsley, [cf. Elements V, 11; and R10]. Those numbers which are the same part or the same parts of the same number or of equal numbers are equal to one another.

Those numbers, which have the same or equal number of the same part or parts are equal to one another. The unity divides any number. Every number divides itself. If two numbers are multiplied, the one measures the product by the other and the latter by the former.

## Pascal, Blaise | Internet Encyclopedia of Philosophy

If a number measures another number by a third one, the third also measures the latter by the former. If a number measures another number by a third one, the product of the first by the third, or of the third by the first, results in the second. It is possible to make additions, subtractions, multiplications, divisions, extractions of roots, and squares and cubes of numbers.

Every property of a number is also a property of a number equal to it. It is the same to multiply a whole with a whole, or a whole with each of the parts, or each of the parts with a whole. Arnauld, [cf. No magnitude has a divisor greater than itself. The greatest common divisor of two parts composing a whole is also the greatest common divisor of one part and the whole. The greatest common divisor of a whole and one of its two parts is also the greatest common divisor of the whole and the other part.

Two numbers are always commensurable the one another. If a number is subtracted from another number, the result is a number. It is the same thing to multiply 12 times 8, or 8 times Elements VII, 16]. It is the same thing to multiply 12 times 8, or 12 times many parts which, taken together, are equal to 8. If equal quantities are multiplied by equal quantities, their products will be equal.

If equal quantities are divided by equal quantities, their quotients will be equal. Given a number, to add a unity to it. Given a number, to subtract a unity from it. If the roots of any two quantities are equal, their squares are also equal. If the squares of any two quantities are equal, their roots are also equal. If unequal quantities are multiplied by equal quantities, their products will be unequal. If unequal quantities are divided by equal quantities, their quotients will be unequal.

Mathematical sciences are not abstracted from natural things; nor are they in the imagination, nor are they in the intellect; but rather, space is their subject. Geometry takes into consideration the point, the line, the surface, and the body, in this natural order. Space is extension, and extension is space. Every space is a minimum, a maximum, or in the middle of these. Every space is long; or long and wide; or long and wide and deep.

Given a thing, to take in it a point or a straight line. Ricci, The object of pure geometry is space, which is considered as extended in three dimensions. Pascal, [cf. S1 and S5]. Space is infinite in each dimension. Pascal, Space is immovable as a whole and its parts are also immovable. Points may only differ in situation. Lines may differ in situation, magnitude and direction. Surfaces may differ in situation, length, breadth, content, and direction. Equal straight lines only differ in situation.

Equal circles only differ in situation. Equal arcs of the same circle only differ in situation. Cords of equal arcs of equal circles only differ in situation. If an end of a straight line is given in position, and the straight line is given in magnitude, the other end of the straight line will fall on the circumference of the circle with that given center and radius. To assume a figure, whose properties are looked for in a theorem, even though it is not constructed. There exists a solid body. A straight line is given in position and magnitude when its endpoints are given. A surface cuts a surface in a line.

If two surfaces which cut one another are planes, they cut one another in a straight line. A line cuts a line in a point. To indefinitely extend any plane. Two straight lines do not have a common segment. Clavius, [from P roclus , In Euclidis ; cf. Two straight lines converging into a point will necessarily intersect one another at that point when produced.

If a point lies on two straight lines, it will be at their intersection or meeting. If two points are on the same plane, the straight line joining them will be on the same plane. Two concurrent and intersecting straight lines have no common part. Grienberger, [another version of I6]. From any point of a circular arc that is not one of its two ends, two straight lines may be drawn to the ends of the arc, forming a triangle with the chord between the ends. Given two points, to draw a circle having one of the points as its center and passing through the other point.

Through any line, to draw a plane. Two planes do not have a common [ plane ] segment. Two planes do not enclose a solid figure. The common section of two planes is a line, and if a line lies on two planes it is their common section. Two intersecting straight lines lie on the same plane. Borelli, [ Elements XI, 2]. Two intersecting straight lines may have at most one point in common. A bounded solid body can be divided into two parts in any direction. Taking together all the divisions, it will be divided into an innumerable number of parts.

The division of a body may be physical, thus disconnecting the parts of the body, or mathematical, thus preserving its continuity. Besides the surfaces that bound a solid body, innumerable other surfaces may be conceived of which bound its parts. A bounded surface can be divided into an innumerable number of lines, and a bounded line into an innumerable number of points. The existence of a bounded solid body implies the existence of surfaces, the existence of bounded surfaces that of lines, the existence of bounded lines that of points.

However big a bounded solid body is posited, a bigger one may be conceived of that contains the former as a proper part. The included body may be conceived of as separated from the bigger one by some space. A solid body revolved around two fixed points has an entire line fixed. Roberval, [cf. If a line is perpendicular to another line, its extension is also perpendicular to it. If a line is perpendicular to another line, the latter is also perpendicular to the former. If a straight line perpendicular to a plane moves uniformly in a straight line on the plane, its flow will produce a plane perpendicular to the first plane.

If a straight line is perpendicular to a plane, a plane passing through the straight line is also perpendicular to the first plane. The extremities of a line are points. Lamy, [ Elements I, def. A straight line is determined by two points only. Only one straight line may be drawn between two points. Lamy, [cf.

Two straight lines that have two points in common coincide. A straight line cannot be partly on one plane and partly on another plane. Lamy, [ Elements XI, 1]. Every triangle lies on a plane. Lamy, [ Elements XI, 2]. If a part of a plane is congruent to a part of another plane, the two planes are congruent to one another. Marchetti, [cf. A straight line [i. Rondelli, To draw a straight line perpendicular to any given straight line in any given point.

Simpson, [ Elements I, 11 and 12]. An infinite straight line divides the infinite plane in two parts. It is possible to extend a plane through two intersecting lines. If a straight or curved line is drawn from a point which is within a figure to another point in the same plane which is outside the figure, it will intersect the sides or boundary of the figure. Fine, If a straight line drawn from any angle of a rectilinear figure meets the opposite side or angle of the figure, it also intersects this latter side or angle. If a circumference with a given center and a radius of a given length is produced, and the radius is extended beyond the center to infinity, it will intersect the circumference.

Only two points on a circumference are on one straight line.

## Umar Khayyam

That is: if a straight line intersects a circle entering in its area and leaving it, it will intersect the circle at two points only, the one when it enters it and the other when it leaves it. If a circumference is described which has an end of a given segment as its center and passes through the other end of the segment, it will also pass through the ends of all segments equal to the given one that are produced from the given center.

If a circumference is described which has its center on the side or boundary of a given figure, and whose radius falls inside the figure not on its boundary , it will intersect the figure at least once. If a circumference is described which has its center outside a given figure, and whose radius falls inside the figure not on its boundary , it will intersect the figure at least once.

If a straight line drawn inside a figure is produced to infinity if needed in both directions, it will intersect the boundary of the figure. If a straight line divides a rectilinear angle and is produced to infinity, it will intersect a given straight line which is applied [ with its ends ] to the straight lines forming the angle.

And a straight line drawn from a rectilinear angle to a point inside its base will divide the angle. And a straight line drawn from a point of the base of a triangle to the opposite angle will divide the angle. If a point is taken inside a space which is everywhere bounded by lines, and an infinite straight line passes through the point, it will cut the boundary of the space at at least two points. If two points are taken on the different sides of a straight line, a straight line going from one point to the other will intersect the first line at exactly one point.

An infinite straight line passing through a point inside a circle intersects the circumference at exactly two points. A circumference passing through a point inside a circle and a point outside it intersects the circumference of the circle at exactly two points. If two circumferences both have some points inside one another, they will intersect one another at exactly two points.

If a circumference has a point on one side of an infinite straight line and the center on the other side, it intersects the straight line at exactly two points. If the same straight line is inside two figures, the two figures will have a common part and will intersect one another. If two equal circles have their centers on the ends of a segment taken as their radius, they meet. Schott, If an infinite straight line passes inside a plane or a solid figure, it will intersect the boundaries of this figure.

If a segment has one end inside a circle and one end outside it, it intersects the circumference at one point. Quantities and their properties may be superposed by an intellectual act. Quantities may be conceived to move completely or around something fixed. Quantities the divisible boundaries of which are congruent are themselves congruent and equal to one another.

Equal straight lines are congruent with one another. Equal rectilinear angles are congruent with one another. Equal circles are congruent with one another. Equilateral and equiangular rectilinear figures are congruent with one another, and equal. If the ends of two straight lines are congruent with one another, the straight lines are themselves congruent and equal. The mere change of position of a triangle or any other plane and rectilinear figure does not change anything in its sides or angles. Given a thing, to reproduce it. Given two things, to superpose them.

If the boundaries of two plane figures are congruent, the two figures are themselves congruent. If two plane figures are congruent, their boundaries are congruent as well. Tacquet, [cf. Equal angles are congruent with one another. Barrow, [cf. T3, and E4]. To draw a straight line equal to a given straight line. Arnauld, [ Elements I, 2]. Things with equal motions traverse equal spaces in equal times. To move the compass keeping its opening. To bend a straight line through the movement of the sides. If a rectilinear angle is congruent with another rectilinear angle and it is reflected without changing its magnitude or width, it will be still congruent with the second angle.

To put two straight lines directly one to the other. Scarburgh, Any extended entity you please may move with any given law whatever. Newton, — A moving body may halt in any given position whatever. Extended entities may describe the paths marked out through the courses of moving things or by the position of stationary ones.

To draw through two given points a line congruent to a given line at a distance no less than that of the points, an ordained point of which shall coincide with one or other given point. To draw any line on which there shall always fall a point which is given according to a precise rule by drawing from points through points lines congruent to given ones. To extend from any given point a straight line equal to a given straight line.

Marchetti, [ Elements I, 2; cf. To cut from any given straight line a portion equal to a shorter given straight line. Marchetti, [ Elements I, 3]. To reproduce any straight line or rectilinear angle.

Simpson, [ Elements I, 2—3 and Elements I, 23]. The magnitudes that are congruent with one another are similar and equal, and the magnitudes that are similar and equal are congruent with one another. The shortest line between two points is the straight line. Bradwardine, fourteenth century [A rchim. De sphaera et cylindro I, Ass. If two more sides are drawn on the base of a triangle, they will be greater or lesser than the previous sides. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal sides equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.

Pelletier, [ Elements I, 4]. If one of several equal angles is a right angle, all angles will be right angles. P roclus , In Euclidis ]. The distance between two points is the straight line. A rchim. The diameter bisects the circle. Pascal, [ Elements I, def. All lines drawn from the center to the circumference of a circle are equal to one another. Two circles that have equal radii are equal. Two equal circles have equal radii. If, from the two ends of a straight line, there is drawn an arc of a circle, as well as two straight lines meeting at a point outside of the circle and not intersecting the circle, and a curve entirely outside the circle, the latter curve and the two straight lines will both be longer than the circle, and the circle will be longer that the initial straight line.

If two lines on the same plane have their ends in common and are both convex toward the same part, the one which is contained by the other is shorter than it. Arnauld, [A rchim. De sphaera et cylindro , Ass.

The degrees of equal circumferences are equal. Among all straight lines drawn from the center of a circle, those smaller than the radius have their other end inside the circle, those equal to the radius have their other end on the circumference, and those longer than the radius have their other end outside of the circle. If two straight lines intersect, and two points of the cutting line are singularly equidistant from two points of the other line, then all other points in the cutting line will also be equidistant from those two points of the other line.

Squares with equal sides are equal to one another. Rectangles with equal bases and heights are equal to one another. Similar segments of circles on equal straight lines are equal to one another. Arnauld, [ Elements III, 24]. A line between two points that deviates toward one part of the other from the straight line between the same points is longer than the straight line. A polygon is larger than the circle inscribed in it.

A polygon is smaller than the circle in which it is inscribed. If one end of a straight line lies on a plane and is conceived as the center of a circle in the plane, the straight line is perpendicular to the plane if and only if the other end is equidistant from the circumference of the circle. A solid figure is bigger than the one to which it is circumscribed and smaller than the one to which it is inscribed.

### Additional information

Among prisms with equal height, the one with the smaller base is smaller. In equal circles, arcs of equal chords are equal and chords of equal arcs are equal. Lamy, [ Elements III, 23 and 24; cf. Equal circles have equal circumferences. If two circles have equal circumferences, they are equal.

## The consolations of understanding

A greater circle has a greater circumference and greater radii. If a circle has a greater circumference or radius than another circle, it is greater than the latter. In a triangle two sides taken together in any manner are greater than the remaining one. Marchetti, [ Elements I, 20]. If two straight lines forming an angle have the same length as two other straight lines forming an equal angle, the segment joining the ends of the former lines will be equal to the segment joining the ends of the latter lines.

Simpson, [ Elements I, 4; cf. If two solids of the same height are cut by planes parallel to their bases and the sections at equal distance from the bases are either equal or in constant ratio, the solids themselves are equal or in the same constant ratio. Arcs of a circle that contain equal angles are similar to one another. Upright prisms with equal bases and heights are equal to one another. Elements XI, 29].

The circumference of a circle is greater than the perimeter of any inscribed polygon, and lesser than the perimeter of any circumscribed polygon. E19 and E20]. If two straight lines lying on the same plane and extended to infinity in both directions do not meet, they are equidistant. If a straight line meets one parallel, it will also meet the other parallels. Parallel lines have a common perpendicular. Tacquet, Two perpendiculars cut off equal segments from each of two parallel lines. If a straight line perpendicular to a given straight line is translated into a plane remaining with one of its ends on the given straight line, the other end of it will draw with its flow another straight line.

If two straight lines extended toward one side come closer and closer, they will eventually meet. If two lines are parallel, all the perpendiculars from one to the other are equal to one another. Dechales, If two straight lines diverge more and more when they are extended in one direction, they will converge and meet in the other direction. Two straight lines do not incline toward one another if one of them does not incline toward a third, unless the other also inclines toward it. Mercator, Two non-equidistant straight lines lying on the same plane, if produced far enough, will meet on one side.

Rondelli, [cf. If two points on a straight line have different distances to another straight line, the two straight lines, infinitely produced, will meet. A straight line passing through a point that is inside an angle cuts at least one of the sides of the angle.

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Lorenz, Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another. Playfair, If two straight lines forming an angle are infinitely produced from a point, their distance will exceed any finite magnitude. If one assumes that two straight lines which form internal angles equal to two right angles when intersected by a third straight line meet in that direction, one should also grant that the two lines always meet when they form the same angles with a third line. For every two points A, B there exists a line a that contains each of the points A, B.

For every two points A, B there exists no more than one line that contains each of the points A, B. There exist at least two points on a line. There exist at least three points that do not lie on a line. For every plane there exists a point which it contains. For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. There exist at least four points which do not lie in a plane. Of any three points on a line there exists no more than one that lies between the other two.

If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC. Every angle is congruent to itself, i. Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a.

Axiom of Archimedes. An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follow from Axioms I—III and from V-1 is impossible.

Axiom of Linear Completeness.