https://www.wilbourhall.org/pdfs/heath/HeathVolI.pdf
PREFACE
The idea may seem quixotic, but it is nevertheless the
author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars. For the mathematician the important consideration is that
the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles,
invented the methods ah initio, and fixed the terminology.
Mathematics in short is a Greek science, whatever new
developments modern analysis has brought or may bring.
The interest of the subject for the classical scholar is no doubt of a different kind. Greek mathematics reveals an
important aspect of the Greek genius of which the student of Greek culture is apt to lose sight. Most people, when they
think of the Greek genius, naturally call to mind its masterpieces in literature and art with their notes of beauty, truth,
freedom and humanism. But the Greek, with his insatiable desire to know the true meaning of everything in the uni- verse and to be able to give a rational explanation of it, was
just as irresistibly driven to natural science, mathematics, and
exact reasoning in general or logic. This austere side of the
Greek genius found perhaps its most complete expression in
Aristotle. Aristotle would, however, by no means admit that
mathematics was divorced from aesthetic ; he could conceive,
he said, of nothing more beautiful than the objects of mathematics. Plato 'delighted in geometry and in the wonders of numbers ; (iyea)fj.irprjTos /J-rjSel^ da-irai, said the inscription over the door of the Academy. Euclid was a no le.ss typical
Gi'eek. Indeed, seeing that so much of Greek is mathematics,
vi PREFACE
it iH arguable that, if one would understand the Greek genius
fully, it Avould be a good plan to begin with their geometry.
The story of Greek mathematics has been written before.
Dr. James Gow did a great service by the publication in 1884
of his Short Hidory of Greek Mathematics, a scholarly and
useful work which has held its own and has been quoted with
respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he
wrote, however, Dr. Gow had necessarily to rely upon the
works of the pioneers Bretschneider, Hankel, AUman, and
Moritz Cantor (first edition). Since then the subject has been
very greatly advanced ; new texts have been published, important new doeumejits have been discovered, and researches
by scholars and mathematicians in different countries have
thrown light on many obscure points. It is, therefore, high
time for the complete story to be r
PREFACE vii
Ulrico Hoepli, Milano). Professor Loria arranges liis material
in five Books, (1) on pre-Euclidean geometry, (2) on tlie Golden Age of Greek geometry (Euclid to Apollonius), (3) on
applied mathematics, including astronomy, sphaeric, optics,
&c., (4) on the Silver Age of Greek geometry, (5) on the
arithmetic of the Greeks. Within the separate Books the
arrangement is chronological, under the names of persons or
schools. I mention these details because they raise the
question whether, in a history of this kind, it is best to follow
chronological order or to arrange the material according to
subjects, and, if the latter, in what sense of the word 'subject'
and within what limits. As Professor Loria says, his arrangement is ' a compromise between arrangement according to subjects and a strict adherence to chronological order, each of which plans has advantages and disadvantages of its own '. In this book I have adopted a new arrangement, mainly
according to subjects, the nature of which and the reasons for which will be made clear by an illustration. Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent,
the finding of two mean proportionals in continued proportion
between tw^o given straight lines. Under a chronological
arrangement this problem comes up afresh on the occasion of
each new solution. Now it is obvious that, if all the recorded
solutions are collected together, it is much easier to see the
relations, amounting in some eases to substantial identity,
between them, and to get a comprehensive view of the history
of the problem. I have therefore dealt with this problem in
a separate section of the chapter devoted to ' Special Problems',
and I have followed the same course with the other famous
problems of squaring the circle and trisecting any angle.
Similar considerations arise with regard to certain welldefined subjects such as conic sections. It would be inconvenient to interrupt the account of Menaechmus's solution
of the problem of the'two mean proportionals in order to
viii PREFACE
consider the way in which he may have discovered the conic
sections and their fundamental properties. It seems to me
much better to give the complete story of the origin and
development of the geometry of the conic sections in one
place, and this has been done in the chapter on conic sections
associated with the name of Apollonius of Perga. Similarly
a chapter has been devoted to algebra (in connexion with
Diophantus) and another to trigonometry (under Hipparchus,
Menelaus and Ptolemy).
At the same time the outstanding personalities of Euclid
and Archimedes demand chapters to themselves. Euclid, the
author of the incomparable Elements, wrote on almost all
the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was
even wider in its range of subjects. The imperishable and
unique monuments of the genius of these two men must be
detached from their surroundings and seen as a whole if we
would appreciate to the full the pre-eminent place which they
occupy, and will hold for all time, in the history of science. The arrangement which I have adopted necessitates (as does
any other order of exposition) a certain amount of repetition
and cross-references ; but only in this way can the necessary
unity be given to the whole narrative.
One other point should be mentioned. It is a defect in the
existing histories that, while they state generally the contents
of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained.
I have therefore taken pains, in the most significant cases,
to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used
and to apply it, if he will, to other similar investigations.
The work was begun in 1913, but the bulk of it was
written, as a distraction, during the first three years of the
PREFACE ix
war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato
to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he
replied, ' It must be supposed, not that the god specially
wished this problem solved, but that he would have the
Greeks desist from war and wckedness and cultivate the
Muses, so that, their passions being assuaged by philosophy
and mathematics, they might live in innocent and mutually
helpful intercourse with one another '. Truly
Greece and her foundations are Built below the tide of war,
Based on the cryst&.lline sea Of thought and its eternity. T. L. H.
Late Greek Period: 200 BC–500 AD
• Apollonius (225 BC): wrote eight-volume book Conics on the subject.
• Greece falls under Roman rule.
• Alexandria remained important: educated Greeks still spoke and wrote greek rather than Latin.
• Hipparchus (140 BC) computed chords (essentially sine tables, although word was not used)
for Astronomy.
• Greek mathematics had something of a hiatus until c.100 AD: this included the time when
Julius Caesar ruled Rome (died 44 BC).
• Heron (75 AD) proved the formula p
s(s − a)(s − b)(s − c) for the area of triangle, where the
semi-perimeter is s = 1
2
(a + b + c).
• Ptolemy (150 AD) extended the work of Hipparchus on trigonometry and wrote the astronomical masterwork Almagest.11
• c.400 Theon and Hypatia produce the most widely-read edition of Euclid’s Elements as well as
improving upon several earlier mathematical topics.
• In 395 the Roman empire split into eastern and western parts, the western rapidly declined under the pressures of corruption and attacks by barabrians. By 500 AD, the western empire had
collapsed. The turmoil did not leave the former parts of the Greek empire untouched. Alexandria experienced riots and a bloody power-struggle (Hypatia herself was murdered by a mob
in 415) and the library of Alexandria was severly damaged and possibly destroyed at this time.
In 642, Alexandria was finally captured by the Islamic caliphate. Most of the material in the
library likely survived by being copied and preserved in various places of learning, generally
in the newly rising caliphate.
PREFACE
The idea may seem quixotic, but it is nevertheless the
author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars. For the mathematician the important consideration is that
the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles,
invented the methods ah initio, and fixed the terminology.
Mathematics in short is a Greek science, whatever new
developments modern analysis has brought or may bring.
The interest of the subject for the classical scholar is no doubt of a different kind. Greek mathematics reveals an
important aspect of the Greek genius of which the student of Greek culture is apt to lose sight. Most people, when they
think of the Greek genius, naturally call to mind its masterpieces in literature and art with their notes of beauty, truth,
freedom and humanism. But the Greek, with his insatiable desire to know the true meaning of everything in the uni- verse and to be able to give a rational explanation of it, was
just as irresistibly driven to natural science, mathematics, and
exact reasoning in general or logic. This austere side of the
Greek genius found perhaps its most complete expression in
Aristotle. Aristotle would, however, by no means admit that
mathematics was divorced from aesthetic ; he could conceive,
he said, of nothing more beautiful than the objects of mathematics. Plato 'delighted in geometry and in the wonders of numbers ; (iyea)fj.irprjTos /J-rjSel^ da-irai, said the inscription over the door of the Academy. Euclid was a no le.ss typical
Gi'eek. Indeed, seeing that so much of Greek is mathematics,
vi PREFACE
it iH arguable that, if one would understand the Greek genius
fully, it Avould be a good plan to begin with their geometry.
The story of Greek mathematics has been written before.
Dr. James Gow did a great service by the publication in 1884
of his Short Hidory of Greek Mathematics, a scholarly and
useful work which has held its own and has been quoted with
respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he
wrote, however, Dr. Gow had necessarily to rely upon the
works of the pioneers Bretschneider, Hankel, AUman, and
Moritz Cantor (first edition). Since then the subject has been
very greatly advanced ; new texts have been published, important new doeumejits have been discovered, and researches
by scholars and mathematicians in different countries have
thrown light on many obscure points. It is, therefore, high
time for the complete story to be r
PREFACE vii
Ulrico Hoepli, Milano). Professor Loria arranges liis material
in five Books, (1) on pre-Euclidean geometry, (2) on tlie Golden Age of Greek geometry (Euclid to Apollonius), (3) on
applied mathematics, including astronomy, sphaeric, optics,
&c., (4) on the Silver Age of Greek geometry, (5) on the
arithmetic of the Greeks. Within the separate Books the
arrangement is chronological, under the names of persons or
schools. I mention these details because they raise the
question whether, in a history of this kind, it is best to follow
chronological order or to arrange the material according to
subjects, and, if the latter, in what sense of the word 'subject'
and within what limits. As Professor Loria says, his arrangement is ' a compromise between arrangement according to subjects and a strict adherence to chronological order, each of which plans has advantages and disadvantages of its own '. In this book I have adopted a new arrangement, mainly
according to subjects, the nature of which and the reasons for which will be made clear by an illustration. Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent,
the finding of two mean proportionals in continued proportion
between tw^o given straight lines. Under a chronological
arrangement this problem comes up afresh on the occasion of
each new solution. Now it is obvious that, if all the recorded
solutions are collected together, it is much easier to see the
relations, amounting in some eases to substantial identity,
between them, and to get a comprehensive view of the history
of the problem. I have therefore dealt with this problem in
a separate section of the chapter devoted to ' Special Problems',
and I have followed the same course with the other famous
problems of squaring the circle and trisecting any angle.
Similar considerations arise with regard to certain welldefined subjects such as conic sections. It would be inconvenient to interrupt the account of Menaechmus's solution
of the problem of the'two mean proportionals in order to
viii PREFACE
consider the way in which he may have discovered the conic
sections and their fundamental properties. It seems to me
much better to give the complete story of the origin and
development of the geometry of the conic sections in one
place, and this has been done in the chapter on conic sections
associated with the name of Apollonius of Perga. Similarly
a chapter has been devoted to algebra (in connexion with
Diophantus) and another to trigonometry (under Hipparchus,
Menelaus and Ptolemy).
At the same time the outstanding personalities of Euclid
and Archimedes demand chapters to themselves. Euclid, the
author of the incomparable Elements, wrote on almost all
the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was
even wider in its range of subjects. The imperishable and
unique monuments of the genius of these two men must be
detached from their surroundings and seen as a whole if we
would appreciate to the full the pre-eminent place which they
occupy, and will hold for all time, in the history of science. The arrangement which I have adopted necessitates (as does
any other order of exposition) a certain amount of repetition
and cross-references ; but only in this way can the necessary
unity be given to the whole narrative.
One other point should be mentioned. It is a defect in the
existing histories that, while they state generally the contents
of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained.
I have therefore taken pains, in the most significant cases,
to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used
and to apply it, if he will, to other similar investigations.
The work was begun in 1913, but the bulk of it was
written, as a distraction, during the first three years of the
PREFACE ix
war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato
to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he
replied, ' It must be supposed, not that the god specially
wished this problem solved, but that he would have the
Greeks desist from war and wckedness and cultivate the
Muses, so that, their passions being assuaged by philosophy
and mathematics, they might live in innocent and mutually
helpful intercourse with one another '. Truly
Greece and her foundations are Built below the tide of war,
Based on the cryst&.lline sea Of thought and its eternity. T. L. H.
Late Greek Period: 200 BC–500 AD
• Apollonius (225 BC): wrote eight-volume book Conics on the subject.
• Greece falls under Roman rule.
• Alexandria remained important: educated Greeks still spoke and wrote greek rather than Latin.
• Hipparchus (140 BC) computed chords (essentially sine tables, although word was not used)
for Astronomy.
• Greek mathematics had something of a hiatus until c.100 AD: this included the time when
Julius Caesar ruled Rome (died 44 BC).
• Heron (75 AD) proved the formula p
s(s − a)(s − b)(s − c) for the area of triangle, where the
semi-perimeter is s = 1
2
(a + b + c).
• Ptolemy (150 AD) extended the work of Hipparchus on trigonometry and wrote the astronomical masterwork Almagest.11
• c.400 Theon and Hypatia produce the most widely-read edition of Euclid’s Elements as well as
improving upon several earlier mathematical topics.
• In 395 the Roman empire split into eastern and western parts, the western rapidly declined under the pressures of corruption and attacks by barabrians. By 500 AD, the western empire had
collapsed. The turmoil did not leave the former parts of the Greek empire untouched. Alexandria experienced riots and a bloody power-struggle (Hypatia herself was murdered by a mob
in 415) and the library of Alexandria was severly damaged and possibly destroyed at this time.
In 642, Alexandria was finally captured by the Islamic caliphate. Most of the material in the
library likely survived by being copied and preserved in various places of learning, generally
in the newly rising caliphate.
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