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'''Pons Asinorum''' ([[Latin]] for "Bridge of Asses") is a term used to refer to a problem  that severely tests the ability of an inexperienced person, and therefore separates the serious and dedicated students from the “asses.” It is said that students are as reluctant to tackle these problems as donkeys (asses) are to cross over a bridge. Once a student is experienced in his field, however, the problem appears relatively simple. The term can be used to refer to a problem that is a stumbling block in any field, or to a problem whose solution seems pointless.
 
  
The term “Bridge of Asses” first came into use during the Middle Ages, and is most commonly applied to a diagram used to help students of logic identify the middle term in a syllogism, or to [[Euclid]]'s fifth proposition in Book 1 of his ''[[Euclid's Elements|Elements]]'' of [[geometry]]. As early as the sixth century, the Greek philosopher Philoponus used a diagram to show what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises.  
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'''Bridge of Asses''' or '''Pons Asinorum''' ([[Latin]] for "Bridge of Asses") is a term used to refer to a problem that severely tests the ability of an inexperienced person, and therefore separates the serious and dedicated students from the “asses.” It is said that students are as reluctant to tackle these problems as [[donkey]]s (asses) are to cross over a bridge. Once a student is experienced in his field, however, the problem appears relatively simple. The term can be used to refer to a problem that is a stumbling block in any field, or to a problem whose solution seems pointless.
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The term “Bridge of Asses” first came into use during the [[Middle Ages]], and is most commonly applied to a diagram used to help students of [[logic]] identify the middle term in a [[syllogism]], or to [[Euclid]]'s fifth proposition in Book 1 of his ''[[Euclid's Elements|Elements]]'' of [[geometry]]. As early as the sixth century, the Greek [[philosophy|philosopher]] [[Philoponus]] used a diagram to show what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises.  
  
== “Pons Asinorum” in Logic==
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== “Pons Asinorum” in [[Logic]]==
  
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The sixth century Greek philosopher [[Philoponus]], presented a diagram showing what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises, to enable students of logic to construct valid syllogisms more easily.<ref>[http://plato.stanford.edu/entries/philoponus/ John Philoponus] Stanford Encyclopedia of Philosophy. Retrieved November 7, 2007.</ref>
  
The sixth century Greek philosopher Philoponus, presented a diagram showing what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises, to enable students of logic to construct valid syllogisms more easily .<ref>[http://plato.stanford.edu/entries/philoponus/ John Philoponus], Stanford Encyclopedia of Philosophy. Retrieved October 21, 2007.</ref>
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The [[France|French]] philosopher Jean Buridan (Joannes Buridanus, c. 1297 &ndash; 1358), professor of philosophy in the University of Paris, is credited with devising a set of rules to help slow-witted students in the discovery of syllogistic middle terms, which later became known as the pons asinorum.  
  
The French philosopher Jean Buridan (Joannes Buridanus, c. 1297-c. 1358), professor of philosophy in the university of Paris, is credited with devising a set of rules to help slow-witted students in the discovery of syllogistic middle terms, which later became known as the pons asinorum.
+
In 1480, Petrus Tartaretus applied the Latin expression “pons asinorum” to a diagram illustrating these rules, whose purpose was to help the student of logic find the middle term of a syllogism and disclose its relations to the other terms.<ref>  
.
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Dagobert D. Runes. Dictionary of Philosophy.</ref>
In 1480, Petrus Tartaretus applied the Latin expression “pons asinorum” to a diagram illustrating these rules, whose purpose was to help the student of logic find the middle term of a syllogism and disclose its relations to the other terms. <ref>  
 
Dagobert D. Runes, Dictionary of Philosophy</ref>
 
  
The “asses’ bridge” was usually presented with the predicate, or major term, of the syllogism on the left, and the subject on the right. The three possible relations of the middle term to either the subject or the predicate (consequent, antecedent and extraneous) were represented by six points arranged in two rows of three in the middle of the diagram, between the subject and the predicate. The student was then asked to identify the nineteen valid combinations of the three figures of the syllogism and evaluate the strength of each premise. <ref>For a more detailed explanation of how the “bridge” works, see C L Hamblin “An Improved Pons Asinorum?” Journal of the History of Philosophy xiv, 1976, pp. 131–36. I</ref> <ref> [http://www.she-philosopher.com/images/gallery/exhibits/Evans20(980x1301).jpg (picture)]. Retrieved October 21, 2007.</ref>
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The “asses’ bridge” was usually presented with the predicate, or major term, of the syllogism on the left, and the subject on the right. The three possible relations of the middle term to either the subject or the predicate (consequent, antecedent and extraneous) were represented by six points arranged in two rows of three in the middle of the diagram, between the subject and the predicate. The student was then asked to identify the nineteen valid combinations of the three figures of the syllogism and evaluate the strength of each premise.<ref>For a more detailed explanation of how the “bridge” works, see C L Hamblin “An Improved Pons Asinorum?” Journal of the History of Philosophy xiv, 1976, pp. 131–36.</ref><ref> [http://www.she-philosopher.com/images/gallery/exhibits/Evans20(980x1301).jpg Picture]. ‘‘She-philosopher.com’’. Retrieved November 7, 2007.</ref>
  
 
==Fifth Proposition of Euclid==
 
==Fifth Proposition of Euclid==
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Euclid's Fifth Proposition reads:
  
{{quote|In isosceles [[triangle|triangles]] the [[angle|angles]] at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Euclid's Fifth Proposition}}
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<blockquote>In isosceles [[triangle|triangles]] the [[angle|angles]] at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.  
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</blockquote>
  
[[Pappus of Alexandria|Pappus]] provided the shortest [[Mathematical proof|proof]] of the first part, that if the triangle is ABC with AB being the same length as AC, then comparing it with the triangle ACB (the [[mirror image]] of triangle ABC) will show that two sides and the included angle at A of one are equal to the corresponding parts of the other, so by the fourth proposition (on congruent triangles) the angles at B and C are equal. The difficulty lies in treating one triangle as two, or in making a correspondence, but not the correspondence of identity, between a triangle and itself.  
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[[Pappus of Alexandria|Pappus]] provided the shortest [[Mathematical proof|proof]] of the first part, that if the triangle is ABC with AB being the same length as AC, then comparing it with the triangle ACB (the [[mirror image]] of triangle ABC) will show that two sides and the included angle at A of one are equal to the corresponding parts of the other, so by the fourth proposition (on congruent triangles) the angles at B and C are equal. The difficulty lies in treating one triangle as two, or in making a correspondence, but not the correspondence of identity, between a triangle and itself.  
 
Euclid's proof was longer and involved the construction of additional triangles:
 
Euclid's proof was longer and involved the construction of additional triangles:
  
<blockquote>'''Proposition 5'''<br>
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<blockquote>'''Proposition 5'''<br/>
 
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
 
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. (Book I.Definition 20; Postulate 2)<br><br>
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Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. (Book I.Definition 20; Postulate 2)<br/><br/>
  
I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE.
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I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE.  
Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. (Book I. Proposition 3.; Postulate.1)<br><br>
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Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. (Book I. Proposition 3.; Postulate.1)<br/><br/>
  
Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG.<br><br>
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Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG.<br/><br/>
 
 
Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. (Book I.Proposition 4)<br><br>
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Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. (Book I.Proposition 4)<br/><br/>
  
Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. (Common Notion 3)<br><br>
+
Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. (Common Notion 3)<br/><br/>
  
But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. (Book I. Proposition 4)<br><br>
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But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. (Book I. Proposition 4)<br/><br/>
  
Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base.(Common Notion 3)<br><br>
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Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base.(Common Notion 3)<br/><br/>
  
Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.<br>
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Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.<br/>
From D.E.Joyce's presentation of Euclid's ''Elements''  [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html]</blockquote>
 
  
 +
From D.E. Joyce's presentation of Euclid's ‘‘Elements’’ <ref>D.E. Joyce. [http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html Euclid's Element Book I, Proposition 5] Retrieved November 7, 2007.</ref></blockquote>
  
It is the ass's pitfall, not his bridge.<br>If this be rightly called the “Bridge of Asses,”<br>He's not the fool who sticks, but he that passes.<br>E. Cobham Brewer, ''Dictionary of Phrase and Fable'', 1894
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<blockquote>It is the ass's pitfall, not his bridge.<br/>If this be rightly called the “Bridge of Asses,”<br/>He's not the fool who sticks, but he that passes.<ref>E. Cobham Brewer. ''Dictionary of Phrase and Fable'', 1894.</ref>
 +
</blockquote>
  
 
==Notes==
 
==Notes==
 
<references/>
 
<references/>
 
==References==
 
==References==
*Boardman, John, Jasper Griffin, and Oswyn Murray. 1986. ''The Oxford history of the classical world''. Oxford: Oxford University Press. ISBN 0198721129 ISBN 9780198721123 ISBN 0192852361 ISBN 9780192852366
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*Boardman, John, Jasper Griffin, and Oswyn Murray. ''The Oxford history of the classical world''. Oxford: Oxford University Press, 1986. ISBN 0198721129
*Broadie, Alexander. 1987. ''Introduction to medieval logic''. Oxford: Clarendon Press. ISBN 0198249411 ISBN 9780198249412
+
*Broadie, Alexander. ''Introduction to medieval logic''. Oxford: Clarendon Press, 1987. ISBN 0198249411
*Euclid. 2005. ''The first three books of Euclid's Elements of geometry from the text of Dr. Robert Simson, together with various useful theorems and problems as geometrical exercises on each book''. Kessinger Publishing's rare mystical reprints. Belle Fourche, S.D.: Kessinger. ISBN 1417945982 ISBN 9781417945986
+
*Euclid. ''The first three books of Euclid's Elements of geometry from the text of Dr. Robert Simson, together with various useful theorems and problems as geometrical exercises on each book''. Kessinger Publishing's rare mystical reprints. Belle Fourche, S.D.: Kissinger, 2005. ISBN 1417945982
*Fauvel, John, and Jeremy Gray. 1987. ''The History of mathematics a reader''. Basingstoke: Macmillan Education in association with the Open University. ISBN 0333427912 ISBN 9780333427910 ISBN 0333427904 ISBN 9780333427903
+
*Fauvel, John, and Jeremy Gray. ''The History of mathematics a reader''. Basingstoke: Macmillan Education in association with the Open University, 1987. ISBN 0333427912
*Magee, Bryan. 1998. ''The story of philosophy''. New York: DK Pub. ISBN 078943511X ISBN 9780789435118
+
*Magee, Bryan. ''The story of philosophy''. New York: DK Pub., 1998. ISBN 078943511X
*Sorabji, Richard. 1987. ''Philoponus and the rejection of Aristotelian science''. Ithaca, N.Y.: Cornell University Press. ISBN 0801420490 ISBN 9780801420498
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*Sorabji, Richard. ''Philoponus and the rejection of Aristotelian science''. Ithaca, N.Y.: Cornell University Press, 1987. ISBN 0801420490
  
==External link==
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==External links==
*[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html D.E.Joyce's presentation of Euclid's ''Elements'']. Retrieved October 21, 2007.
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All links retrieved February 11, 2022.
[[Category:History of mathematics]]
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*[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html D.E.Joyce's presentation of Euclid's ''Elements'']  
[[Category:Elementary geometry]]
 
[[Category:Triangle geometry]]
 
  
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===General Philosophy Sources===
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*[http://plato.stanford.edu/ Stanford Encyclopedia of Philosophy]
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*[http://www.iep.utm.edu/ The Internet Encyclopedia of Philosophy]
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*[http://www.bu.edu/wcp/PaidArch.html Paideia Project Online]
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*[http://www.gutenberg.org/ Project Gutenberg]
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 +
[[Category:mathematics]]
 +
[[Category:philosophy]]
 
[[Category:philosophy and religion]]
 
[[Category:philosophy and religion]]
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{{credits|Pons_asinorum|153281742}}

Revision as of 13:32, 11 February 2022


Bridge of Asses or Pons Asinorum (Latin for "Bridge of Asses") is a term used to refer to a problem that severely tests the ability of an inexperienced person, and therefore separates the serious and dedicated students from the “asses.” It is said that students are as reluctant to tackle these problems as donkeys (asses) are to cross over a bridge. Once a student is experienced in his field, however, the problem appears relatively simple. The term can be used to refer to a problem that is a stumbling block in any field, or to a problem whose solution seems pointless.

The term “Bridge of Asses” first came into use during the Middle Ages, and is most commonly applied to a diagram used to help students of logic identify the middle term in a syllogism, or to Euclid's fifth proposition in Book 1 of his Elements of geometry. As early as the sixth century, the Greek philosopher Philoponus used a diagram to show what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises.

“Pons Asinorum” in Logic

The sixth century Greek philosopher Philoponus, presented a diagram showing what kind of conclusions (universal affirmative, universal negative, particular affirmative, or particular negative) follow from what kind of premises, to enable students of logic to construct valid syllogisms more easily.[1]

The French philosopher Jean Buridan (Joannes Buridanus, c. 1297 – 1358), professor of philosophy in the University of Paris, is credited with devising a set of rules to help slow-witted students in the discovery of syllogistic middle terms, which later became known as the pons asinorum.

In 1480, Petrus Tartaretus applied the Latin expression “pons asinorum” to a diagram illustrating these rules, whose purpose was to help the student of logic find the middle term of a syllogism and disclose its relations to the other terms.[2]

The “asses’ bridge” was usually presented with the predicate, or major term, of the syllogism on the left, and the subject on the right. The three possible relations of the middle term to either the subject or the predicate (consequent, antecedent and extraneous) were represented by six points arranged in two rows of three in the middle of the diagram, between the subject and the predicate. The student was then asked to identify the nineteen valid combinations of the three figures of the syllogism and evaluate the strength of each premise.[3][4]

Fifth Proposition of Euclid

Euclid's Fifth Proposition reads:

In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

Pappus provided the shortest proof of the first part, that if the triangle is ABC with AB being the same length as AC, then comparing it with the triangle ACB (the mirror image of triangle ABC) will show that two sides and the included angle at A of one are equal to the corresponding parts of the other, so by the fourth proposition (on congruent triangles) the angles at B and C are equal. The difficulty lies in treating one triangle as two, or in making a correspondence, but not the correspondence of identity, between a triangle and itself. Euclid's proof was longer and involved the construction of additional triangles:

Proposition 5

In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. Let ABC be an isosceles triangle having the side AB equal to the side AC, and let the straight lines BD and CE be produced further in a straight line with AB and AC. (Book I.Definition 20; Postulate 2)

I say that the angle ABC equals the angle ACB, and the angle CBD equals the angle BCE. Take an arbitrary point F on BD. Cut off AG from AE the greater equal to AF the less, and join the straight lines FC and GB. (Book I. Proposition 3.; Postulate.1)

Since AF equals AG, and AB equals AC, therefore the two sides FA and AC equal the two sides GA and AB, respectively, and they contain a common angle, the angle FAG.

Therefore the base FC equals the base GB, the triangle AFC equals the triangle AGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides, that is, the angle ACF equals the angle ABG, and the angle AFC equals the angle AGB. (Book I.Proposition 4)

Since the whole AF equals the whole AG, and in these AB equals AC, therefore the remainder BF equals the remainder CG. (Common Notion 3)

But FC was also proved equal to GB, therefore the two sides BF and FC equal the two sides CG and GB respectively, and the angle BFC equals the angle CGB, while the base BC is common to them. Therefore the triangle BFC also equals the triangle CGB, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. Therefore the angle FBC equals the angle GCB, and the angle BCF equals the angle CBG. (Book I. Proposition 4)

Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG equals the angle BCF, the remaining angle ABC equals the remaining angle ACB, and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB, and they are under the base.(Common Notion 3)

Therefore in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

From D.E. Joyce's presentation of Euclid's ‘‘Elements’’ [5]

It is the ass's pitfall, not his bridge.
If this be rightly called the “Bridge of Asses,”
He's not the fool who sticks, but he that passes.[6]

Notes

  1. John Philoponus Stanford Encyclopedia of Philosophy. Retrieved November 7, 2007.
  2. Dagobert D. Runes. Dictionary of Philosophy.
  3. For a more detailed explanation of how the “bridge” works, see C L Hamblin “An Improved Pons Asinorum?” Journal of the History of Philosophy xiv, 1976, pp. 131–36.
  4. Picture. ‘‘She-philosopher.com’’. Retrieved November 7, 2007.
  5. D.E. Joyce. Euclid's Element Book I, Proposition 5 Retrieved November 7, 2007.
  6. E. Cobham Brewer. Dictionary of Phrase and Fable, 1894.

References
ISBN links support NWE through referral fees

  • Boardman, John, Jasper Griffin, and Oswyn Murray. The Oxford history of the classical world. Oxford: Oxford University Press, 1986. ISBN 0198721129
  • Broadie, Alexander. Introduction to medieval logic. Oxford: Clarendon Press, 1987. ISBN 0198249411
  • Euclid. The first three books of Euclid's Elements of geometry from the text of Dr. Robert Simson, together with various useful theorems and problems as geometrical exercises on each book. Kessinger Publishing's rare mystical reprints. Belle Fourche, S.D.: Kissinger, 2005. ISBN 1417945982
  • Fauvel, John, and Jeremy Gray. The History of mathematics a reader. Basingstoke: Macmillan Education in association with the Open University, 1987. ISBN 0333427912
  • Magee, Bryan. The story of philosophy. New York: DK Pub., 1998. ISBN 078943511X
  • Sorabji, Richard. Philoponus and the rejection of Aristotelian science. Ithaca, N.Y.: Cornell University Press, 1987. ISBN 0801420490

External links

All links retrieved February 11, 2022.

General Philosophy Sources

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