It has been well said that, if we would study any subject properly, we
must study it as something that is alive and growing and consider it
with reference to its growth in the past. As most of the vital forces
and movements in modern civilization had their origin in Greece, this
means that, to study them properly, we must get back to Greece. So it is
with the literature of modern countries, or their philosophy, or their
art; we cannot study them with the determination to get to the bottom
and understand them without the way pointing eventually back to Greece.
When we think of the debt which mankind owes to the Greeks, we are apt
to think too exclusively of the masterpieces in literature and art which
they have left us. But the Greek genius was many-sided; the Greek, with
his insatiable love of knowledge, his determination to see things as
they are and to see them whole, his burning desire to be able to give a
rational explanation of everything in heaven and earth, was just as
irresistibly driven to natural science, mathematics, and exact reasoning
in general, or logic.
To quote from a brilliant review of a well-known work: 'To be a Greek
was to seek to know, to know the primordial substance of matter, to know
the meaning of number, to know the world as a rational whole. In no
spirit of paradox one may say that Euclid is the most typical Greek: he
would know to the bottom, and know as a rational system, the laws of the
measurement of the earth. Plato, too, loved geometry and the wonders of
numbers; he was essentially Greek because he was essentially
mathematical.... And if one thus finds the Greek genius in Euclid and
the _Posterior Analytics_, one will understand the motto written over
the Academy, μηδεις αγεωμετρητος εισιτω {mêdeis ageômetrêtos eisitô}. To
know what the Greek genius meant you must (if one may speak εν αινιγματι
{en ainigmati}) begin with geometry.'
Mathematics, indeed, plays an important part in Greek philosophy: there
are, for example, many passages in Plato and Aristotle for the
interpretation of which some knowledge of the technique of Greek
mathematics is the first essential. Hence it should be part of the
equipment of every classical student that he should have read
substantial portions of the works of the Greek mathematicians in the
original, say, some of the early books of Euclid in full and the
definitions (at least) of the other books, as well as selections from
other writers. Von Wilamowitz-Moellendorff has included in his
_Griechisches Lesebuch_ extracts from Euclid, Archimedes and Heron of
Alexandria; and the example should be followed in this country.
Acquaintance with the original works of the Greek mathematicians is no
less necessary for any mathematician worthy of the name. Mathematics is
a Greek science. So far as pure geometry is concerned, the
mathematician's technical equipment is almost wholly Greek. The Greeks
laid down the principles, fixed the terminology and invented the methods
_ab initio_; moreover, they did this with such certainty that in the
centuries which have since elapsed there has been no need to
reconstruct, still less to reject as unsound, any essential part of
their doctrine.
Consider first the terminology of mathematics. Almost all the standard
terms are Greek or Latin translations from the Greek, and, although the
mathematician may be taught their meaning without knowing Greek, he will
certainly grasp their significance better if he knows them as they arise
and as part of the living language of the men who invented them. Take
the word _isosceles_; a schoolboy can be shown what an isosceles
triangle is, but, if he knows nothing of the derivation, he will wonder
why such an apparently outlandish term should be necessary to express so
simple an idea. But if the mere appearance of the word shows him that it
means a thing _with equal legs_, being compounded of ισος {isos}, equal,
and σκελος {skelos}, a leg, he will understand its appropriateness and
will have no difficulty in remembering it. _Equilateral_, on the other
hand, is borrowed from the Latin, but it is merely the Latin translation
of the Greek ισοπλευρος {isopleuros}, _equal-sided_. _Parallelogram_
again can be explained to a Greekless person, but it will be far better
understood by one who sees in it the two words παραλληλος {parallêlos}
and γραμμη {grammê} and realizes that it is a short way of expressing
that the figure in question is contained by parallel lines; and we shall
best understand the word _parallel_ itself if we see in it the statement
of the fact that the two straight lines so described go _alongside one
another_, παρ' αλληλας {par' allêlas}, all the way. Similarly a
mathematician should know that a _rhombus_ is so called from its
resemblance to a form of spinning-top (ῥομβος {rhombos} from ῥεμβω
{rhembô}, to spin) and that, just as a parallelogram is a figure formed
by two pairs of parallel straight lines, so a _parallelepiped_ is a
solid figure bounded by three pairs of parallel planes (παραλληλος
{parallêlos}, parallel, and επιπεδος {epipedos}, plane); incidentally,
in the latter case, he will be saved from writing 'parallel_o_piped', a
monstrosity which has disfigured not a few textbooks of geometry.
Another good example is the word _hypotenuse_; it comes from the verb
ὑποτεινειν {hypoteinein} (c. ὑπο {hypo} and acc. or simple acc.), to
_stretch under_, or, in its Latin form, to _subtend_, which term is used
quite generally for 'to be opposite to'; in our phraseology the word
_hypotenuse_ is restricted to that side of a right-angled triangle which
is opposite to the right angle, being short for the expression used in
Eucl. i. 47, ἡ την ορθην γωνιαν ὑποτεινουσα πλευρα {hê tên orthên gônian
hypoteinousa pleura}, 'the side subtending the right angle', which
accounts for the feminine participial form ὑποτεινουσα {hypoteinousa},
_hypotenuse_. If mathematicians had had more Greek, perhaps the
misspelt form 'hypot_h_enuse' would not have survived so long.
To take an example outside the Elements, how can a mathematician
properly understand the term _latus rectum_ used in conic sections
unless he has seen it in Apollonius as the _erect side_ (ορθια πλευρα
{orthia pleura}) of a certain rectangle in the case of each of the three
conics?[3] The word _ordinate_ can hardly convey anything to one who
does not know that it is what Apollonius describes as 'the straight line
drawn down (from a point on the curve) in the _prescribed_ or _ordained_
manner (τεταγμενως κατηγμενη {tetagmenôs katêgmenê})'. _Asymptote_ again
comes from ασυμπτωτος {asymptôtos}, _non-meeting_, _non-secant_, and had
with the Greeks a more general signification as well as the narrower one
which it has for us: it was sometimes used of parallel lines, which also
'do not meet'.
[3] In the case of the parabola, the base (as distinct from the
'erect side') of the rectangle is what is called the _abscissa_ (Gk.
αποτεμνομενη {apotemnomenê}, 'cut off') of the ordinate, and the
rectangle itself is equal to the square on the ordinate. In the case
of the central conics, the base of the rectangle is 'the transverse
side of the figure' or the transverse diameter (the diameter of
reference), and the rectangle is equal to the square on the diameter
conjugate to the diameter of reference.
Again, if we take up a textbook of geometry written in accordance with
the most modern Education Board circular or University syllabus, we
shall find that the phraseology used (except where made more colloquial
and less scientific) is almost all pure Greek. The Greek tongue was
extraordinarily well adapted as a vehicle of scientific thought. One of
the characteristics of Euclid's language which his commentator Proclus
is most fond of emphasizing is its marvellous _exactness_ (ακριβεια
{akribeia}). The language of the Greek geometers is also wonderfully
concise, notwithstanding all appearances to the contrary. One of the
complaints often made against Euclid is that he is 'diffuse'. Yet
(apart from abbreviations in writing) it will be found that the
exposition of corresponding matters in modern elementary textbooks
generally takes up, not less, but more space. And, to say nothing of
the perfect finish of Archimedes's treatises, we shall find in Heron,
Ptolemy and Pappus veritable models of concise statement. The purely
geometrical proof by Heron of the formula for the area of a triangle,
Δ{D}=√_{s(s-a)(s-b)(s-c)}_, and the geometrical propositions in Book I
of Ptolemy's _Syntaxis_ (including 'Ptolemy's Theorem') are cases in
point.
The principles of geometry and arithmetic (in the sense of the theory of
numbers) are stated in the preliminary matter of Books I and VII of
Euclid. But Euclid was not their discoverer; they were gradually evolved
from the time of Pythagoras onwards. Aristotle is clear about the nature
of the principles and their classification. Every demonstrative science,
he says, has to do with three things, the subject-matter, the things
proved, and the things from which the proof starts (εξ ὡν {ex hôn}). It
is not everything that can be proved, otherwise the chain of proof would
be endless; you must begin somewhere, and you must start with things
admitted but indemonstrable. These are, first, principles common to all
sciences which are called _axioms_ or _common opinions_, as that 'of two
contradictories one must be true', or 'if equals be subtracted from
equals, the remainders are equal'; secondly, principles peculiar to the
subject-matter of the particular science, say geometry. First among the
latter principles are definitions; there must be agreement as to what we
mean by certain terms. But a definition asserts nothing about the
existence or non-existence of the thing defined. The existence of the
various things defined has to be _proved_ except in the case of a few
primary things in each science the existence of which is indemonstrable
and must be _assumed_ among the first principles of the science; thus in
geometry we must assume the existence of points and lines, and in
arithmetic of the unit. Lastly, we must assume certain other things
which are less obvious and cannot be proved but yet have to be
accepted; these are called _postulates_, because they make a demand on
the faith of the learner. Euclid's Postulates are of this kind,
especially that known as the parallel-postulate.
The methods of solution of problems were no doubt first applied in
particular cases and then gradually systematized; the technical terms
for them were probably invented later, after the methods themselves had
become established.
One method of solution was the _reduction_ of one problem to another.
This was called απαγωγη {apagôgê}, a term which seems to occur first in
Aristotle. But instances of such reduction occurred long before.
Hippocrates of Chios reduced the problem of duplicating the cube to that
of finding two mean proportionals in continued proportion between two
straight lines, that is, he showed that, if the latter problem could be
solved, the former was thereby solved also; and it is probable that
there were still earlier cases in the Pythagorean geometry.
Next there is the method of mathematical _analysis_. This method is said
to have been 'communicated' or 'explained' by Plato to Leodamas of
Thasos; but, like reduction (to which it is closely akin), analysis in
the mathematical sense must have been in use much earlier. _Analysis_
and its correlative _synthesis_ are defined by Pappus: 'in analysis we
assume that which is sought as if it were already done, and we inquire
what it is from which this results, and again what is the antecedent
cause of the latter, and so on, until by so retracing our steps we come
upon something already known or belonging to the class of principles.
But in synthesis, reversing the process, we take as already done that
which was last arrived at in the analysis, and, by arranging in their
natural order as consequences what were before antecedents and
successively connecting them one with another, we arrive finally at the
construction of that which was sought.'
The method of _reductio ad absurdum_ is a variety of analysis. Starting
from a hypothesis, namely the contradictory of what we desire to prove,
we use the same process of analysis, carrying it back until we arrive at
something admittedly false or absurd. Aristotle describes this method in
various ways as _reductio ad absurdum_, proof _per impossibile_, or
proof leading to the impossible. But here again, though the term was
new, the method was not. The paradoxes of Zeno are classical instances.
Lastly, the Greeks established the form of exposition which still
governs geometrical work, simply because it is dictated by strict logic.
It is seen in Euclid's propositions, with their separate formal
divisions, to which specific names were afterwards assigned, (1) the
_enunciation_ (προτασις {protasis}), (2) the _setting-out_ (εκθεσις
{ekthesis}), (3) the διορισμος {diorismos}, being a re-statement of what
we are required to do or prove, not in general terms (as in the
_enunciation_), but with reference to the particular data contained in
the _setting-out_, (4) the _construction_ (κατασκευη {kataskeuê}), (5)
the _proof_ (αποδειξις {apodeixis}), (6) the _conclusion_ (συμπερασμα
{symperasma}). In the case of a problem it often happens that a solution
is not possible unless the particular data are such as to satisfy
certain conditions; in this case there is yet another constituent part
in the proposition, namely the statement of the conditions or limits of
possibility, which was called by the same name διορισμος {diorismos},
definition or delimitation, as that applied to the third constituent
part of a theorem.
We have so far endeavoured to indicate generally the finality and the
abiding value of the work done by the creators of mathematical science.
It remains to summarize, as briefly as possible, the history of Greek
mathematics according to periods and subjects.
The Greeks of course took what they could in the shape of elementary
facts in geometry and astronomy from the Egyptians and Babylonians. But
some of the essential characteristics of the Greek genius assert
themselves even in their borrowings from these or other sources. Here,
as everywhere else, we see their directness and concentration; they
always knew what they wanted, and they had an unerring instinct for
taking only what was worth having and rejecting the rest. This is
illustrated by the story of Pythagoras's travels. He consorted with
priests and prophets and was initiated into the religious rites
practised in different places, not out of religious enthusiasm 'as you
might think' (says our informant), but in order that he might not
overlook any fragment of knowledge worth acquiring that might lie hidden
in the mysteries of divine worship.
This story also illustrates an important advantage which the Greeks had
over the Egyptians and Babylonians. In those countries science, such as
it was, was the monopoly of the priests; and, where this is the case,
the first steps in science are apt to prove the last also, because the
scientific results attained tend to become involved in religious
prescriptions and routine observances, and so to end in a collection of
lifeless formulae. Fortunately for the Greeks, they had no organized
priesthood; untrammelled by prescription, traditional dogmas or
superstition, they could give their reasoning faculties free play. Thus
they were able to create science as a living thing susceptible of
development without limit.
Greek geometry, as also Greek astronomy, begins with Thales (about
624-547 B. C.), who travelled in Egypt and is said to have brought
geometry from thence. Such geometry as there was in Egypt arose out of
practical needs. Revenue was raised by the taxation of landed property,
and its assessment depended on the accurate fixing of the boundaries of
the various holdings. When these were removed by the periodical flooding
due to the rising of the Nile, it was necessary to replace them, or to
determine the taxable area independently of them, by an art of
land-surveying. We conclude from the Papyrus Rhind (say 1700 B. C.) and
other documents that Egyptian geometry consisted mainly of practical
rules for measuring, with more or less accuracy, (1) such areas as
squares, triangles, trapezia, and circles, (2) the solid content of
measures of corn, &c., of different shapes. The Egyptians also
constructed pyramids of a certain slope by means of arithmetical
calculations based on a certain ratio, _se-qeṭ_, namely the ratio of
half the side of the base to the height, which is in fact equivalent to
the co-tangent of the angle of slope. The use of this ratio implies the
notion of similarity of figures, especially triangles. The Egyptians
knew, too, that a triangle with its sides in the ratio of the numbers 3,
4, 5 is right-angled, and used the fact as a means of drawing right
angles. But there is no sign that they knew the general property of a
right-angled triangle (= Eucl. I. 47), of which this is a particular
case, or that they proved any general theorem in geometry.
No doubt Thales, when he was in Egypt, would see diagrams drawn to
illustrate the rules for the measurement of circles and other plane
figures, and these diagrams would suggest to him certain similarities
and congruences which would set him thinking whether there were not some
elementary general principles underlying the construction and relations
of different figures and parts of figures. This would be in accord with
the Greek instinct for generalization and their wish to be able to
account for everything on rational principles.
The following theorems are attributed to Thales: (1) that a circle is
bisected by any diameter (Eucl. I, Def. 17), (2) that the angles at the
base of an isosceles triangle are equal (Eucl. I. 5), (3) that, if two
straight lines cut one another, the vertically opposite angles are equal
(Eucl. I. 15), (4) that, if two triangles have two angles and one side
respectively equal, the triangles are equal in all respects (Eucl. I.
26). He is said (5) to have been the first to inscribe a right-angled
triangle in a circle, which must mean that he was the first to discover
that the angle in a semicircle is a right angle (cf. Eucl. III. 31).
Elementary as these things are, they represent a new departure of a
momentous kind, being the first steps towards a _theory_ of geometry. On
this point we cannot do better than quote some remarks from Kant's
preface to the second edition of his _Kritik der reinen Vernunft_.
'Mathematics has, from the earliest times to which the history of human
reason goes back, (that is to say) with that wonderful people the
Greeks, travelled the safe road of a _science_. But it must not be
supposed that it was as easy for mathematics as it was for logic, where
reason is concerned with itself alone, to find, or rather to build for
itself, that royal road. I believe on the contrary that with mathematics
it remained for long a case of groping about--the Egyptians in
particular were still at that stage--and that this transformation must
be ascribed to a _revolution_ brought about by the happy inspiration of
one man in trying an experiment, from which point onward the road that
must be taken could no longer be missed, and the safe way of a science
was struck and traced out for all time and to distances illimitable....
A light broke on the first man who demonstrated the property of the
isosceles triangle (whether his name was Thales or what you will)....'
Thales also solved two problems of a practical kind: (1) he showed how
to measure the distance of a ship at sea, and (2) he found the heights
of pyramids by means of the shadows thrown on the ground by the pyramid
and by a stick of known length at the same moment; one account says that
he chose the time when the lengths of the stick and of its shadow were
equal, but in either case he argued by similarity of triangles.
In astronomy Thales predicted a solar eclipse which was probably that of
the 28th May 585 B. C. Now the Babylonians, as the result of
observations continued through centuries, had discovered the period of
223 lunations after which eclipses recur. It is most likely therefore
that Thales had heard of this period, and that his prediction was based
upon it. He is further said to have used the Little Bear for finding the
pole, to have discovered the inequality of the four astronomical
seasons, and to have written works _On the Equinox_ and _On the
Solstice_.
After Thales come the Pythagoreans. Of the Pythagoreans Aristotle says
that they applied themselves to the study of mathematics and were the
first to advance that science, going so far as to find in the principles
of mathematics the principles of all existing things. Of Pythagoras
himself we are told that he attached supreme importance to the study of
arithmetic, advancing it and taking it out of the region of practical
utility, and again that he transformed the study of geometry into a
liberal education, examining the principles of the science from the
beginning.
The very word μαθηματα {mathêmata}, which originally meant 'subjects of
instruction' generally, is said to have been first appropriated to
mathematics by the Pythagoreans.
In saying that arithmetic began with Pythagoras we have to distinguish
between the uses of that word then and now. Αριθμητικη {Arithmêtikê}
with the Greeks was distinguished from λογιστικη {logistikê}, the
science of calculation. It is the latter word which would cover
arithmetic in our sense, or practical calculation; the term αριθμητικη
{arithmêtikê} was restricted to the science of numbers considered in
themselves, or, as we should say, the Theory of Numbers. Another way of
putting the distinction was to say that αριθμητικη {arithmêtikê} dealt
with absolute numbers or numbers in the abstract, and λογιστικη
{logistikê} with numbered _things_ or concrete numbers; thus λογιστικη
{logistikê} included simple problems about numbers of apples, bowls, or
objects generally, such as are found in the Greek Anthology and
sometimes involve simple algebraical equations.
The Theory of Numbers then began with Pythagoras (about 572-497 B. C.).
It included definitions of the unit and of number, and the
classification and definitions of the various classes of numbers, odd,
even, prime, composite, and sub-divisions of these such as odd-even,
even-times-even, &c. Again there were figured numbers, namely,
triangular numbers, squares, oblong numbers, polygonal numbers
(pentagons, hexagons, &c.) corresponding respectively to plane figures,
and pyramidal numbers, cubes, parallelepipeds, &c., corresponding to
solid figures in geometry. The treatment was mostly geometrical, the
numbers being represented by dots filling up geometrical figures of the
various kinds. The laws of formation of the various figured numbers were
established. In this investigation the _gnomon_ played an important
part. Originally meaning the upright needle of a sun-dial, the term was
next used for a figure like a carpenter's square, and then was applied
to a figure of that shape put round two sides of a square and making up
a larger square. The arithmetical application of the term was similar.
If we represent a unit by one dot and put round it three dots in such a
way that the four form the corners of a square, _three_ is the first
gnomon. _Five_ dots put at equal distances round two sides of the square
containing four dots make up the next square (3²), and _five_ is the
second gnomon. Generally, if we have _n²_ dots so arranged as to fill up
a square with _n_ for its side, the gnomon to be put round it to make up
the next square, _(n+1)²_, has _2n+1_ dots. In the formation of squares,
therefore, the successive gnomons are the series of odd numbers
following 1 (the first square), namely 3, 5, 7, ... In the formation of
_oblong_ numbers (numbers of the form _n(n+1)_), the first of which is
1. 2, the successive gnomons are the terms after 2 in the series of
_even_ numbers 2, 4, 6.... Triangular numbers are formed by adding to 1
(the first triangle) the terms after 1 in the series of natural numbers
1, 2, 3 ...; these are therefore the gnomons (by analogy) for triangles.
The gnomons for pentagonal numbers are the terms after 1 in the
arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common
difference) and so on; the common difference of the successive gnomons
for an _a_-gonal number is _a-2_.
From the series of gnomons for squares we easily deduce a formula for
finding square numbers which are the sum of two squares. For, the gnomon
_2n+1_ being the difference between the successive squares _n²_ and
_(n+1)²_, we have only to make _2n+1_ a square. Suppose that _2n+1=m²_;
therefore _n=½(m²-1)_, and _{½(m²-1)}²+m²={½(m²+1)}²_, where _m_ is any
odd number. This is the formula actually attributed to Pythagoras.
Pythagoras is said to have discovered the theory of proportionals or
proportion. This was a numerical theory and therefore was applicable to
commensurable magnitudes only; it was no doubt somewhat on the lines of
Euclid, Book VII. Connected with the theory of proportion was that of
_means_, and Pythagoras was acquainted with three of these, the
arithmetic, geometric, and sub-contrary (afterwards called harmonic). In
particular Pythagoras is said to have introduced from Babylon into
Greece the 'most perfect' proportion, namely:
_a:(a+b)/2=2ab/(a+b):b_,
where the second and third terms are respectively the arithmetic and
harmonic mean between _a_ and _b_. A particular case is 12:9=8:6.
This bears upon what was probably Pythagoras's greatest discovery,
namely that the musical intervals correspond to certain arithmetical
ratios between lengths of string at the same tension, the octave
corresponding to the ratio 2:1, the fifth to 3:2 and the fourth to 4:3.
These ratios being the same as those of 12 to 6, 8, 9 respectively, we
can understand how the third term, 8, in the above proportion came to
be called the 'harmonic' mean between 12 and 6.
The Pythagorean arithmetic as a whole, with the developments made after
the time of Pythagoras himself, is mainly known to us through
Nicomachus's _Introductio arithmetica_, Iamblichus's commentary on the
same, and Theon of Smyrna's work _Expositio rerum mathematicarum ad
legendum Platonem utilium_. The things in these books most deserving of
notice are the following.
First, there is the description of a 'perfect' number (a number which is
equal to the sum of all its parts, i.e. all its integral divisors
including 1 but excluding the number itself), with a statement of the
property that all such numbers end in 6 or 8. Four such numbers, namely
6, 28, 496, 8128, were known to Nicomachus. The law of formation for
such numbers is first found in Eucl. IX. 36 proving that, if the sum
(S_{n}) of _n_ terms of the series 1, 2, 2², 2³ ... is prime, then
S_{n}.2^{n-1} is a perfect number.
Secondly, Theon of Smyrna gives the law of formation of the series of
'side-' and 'diameter-' numbers which satisfy the equations _2x²-y²=±1_.
The law depends on the proposition proved in Eucl. II. 10 to the effect
that _(2x+y)²-2(x+y)²=2x²-y²_, whence it follows that, if _x_, _y_
satisfy either of the above equations, then _2x+y_, _x+y_ is a solution
in higher numbers of the other equation. The successive solutions give
values for _y/x_, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are
successive approximations to the value of √2 (the ratio of the diagonal
of a square to its side). The occasion for this method of approximation
to √2 (which can be carried as far as we please) was the discovery by
the Pythagoreans of the incommensurable or irrational in this particular
case.
Thirdly, Iamblichus mentions a discovery by Thymaridas, a Pythagorean
not later than Plato's time, called the επανθημα {epanthêma} ('bloom')
of Thymaridas, and amounting to the solution of any number of
simultaneous equations of the following form:
_x+x₁+x₂+ ... +x_{n-1} = s_,
_x+x₁ = a₁_,
_x+x₂ = a₂_,
...
_x+x_{n-1} = a_{n-1}_,
the solution being _x=((a₁+a₂+ ... +a_{n-1})-s)/(n-2)_.
The rule is stated in general terms, but the above representation of its
effect shows that it is a piece of pure algebra.
The Pythagorean contributions to geometry were even more remarkable. The
most famous proposition attributed to Pythagoras himself is of course
the theorem of Eucl. I. 47 that the square on the hypotenuse of any
right-angled triangle is equal to the sum of the squares on the other
two sides. But Proclus also attributes to him, besides the theory of
proportionals, the construction of the 'cosmic figures', the five
regular solids.
One of the said solids, the dodecahedron, has twelve regular pentagons
for faces, and the construction of a regular pentagon involves the
cutting of a straight line 'in extreme and mean ratio' (Eucl. II. 11 and
VI. 30), which is a particular case of the method known as the
_application of areas_. This method was fully worked out by the
Pythagoreans and proved one of the most powerful in all Greek geometry.
The most elementary case appears in Eucl. I. 44, 45, where it is shown
how to apply to a given straight line as base a parallelogram with one
angle equal to a given angle and equal in area to any given rectilineal
figure; this construction is the geometrical equivalent of arithmetical
_division_. The general case is that in which the parallelogram, though
applied to the straight line, overlaps it or falls short of it in such a
way that the part of the parallelogram which extends beyond or falls
short of the parallelogram of the same angle and breadth on the given
straight line itself (exactly) as base is similar to any given
parallelogram (Eucl. VI. 28, 29). This is the geometrical equivalent of
the solution of the most general form of quadratic equation _ax±mx²=C_,
so far as it has real roots; the condition that the roots may be real
was also worked out (=Eucl. VI. 27). It is in the form of 'application
of areas' that Apollonius obtains the fundamental property of each of
the conic sections, and, as we shall see, it is from the terminology of
application of areas that Apollonius took the three names _parabola_,
_hyperbola_, and _ellipse_ which he was the first to give to the three
curves.
Another problem solved by the Pythagoreans was that of drawing a
rectilineal figure which shall be equal in area to one given rectilineal
figure and similar to another. Plutarch mentions a doubt whether it was
this problem or the theorem of Eucl. I. 47 on the strength of which
Pythagoras was said to have sacrificed an ox.
The main particular applications of the theorem of the square on the
hypotenuse, e. g. those in Euclid, Book II, were also Pythagorean; the
construction of a square equal to a given rectangle (Eucl. II. 14) is
one of them, and corresponds to the solution of the pure quadratic
equation _x²=ab_.
The Pythagoreans knew the properties of parallels and proved the theorem
that the sum of the three angles of any triangle is equal to two right
angles.
As we have seen, the Pythagorean theory of proportion, being numerical,
was inadequate in that it did not apply to incommensurable magnitudes;
but, with this qualification, we may say that the Pythagorean geometry
covered the bulk of the subject-matter of Books I, II, IV and VI of
Euclid's _Elements_. The case is less clear with regard to Book III of
the _Elements_; but, as the main propositions of that Book were known to
Hippocrates of Chios in the second half of the fifth century B. C., we
conclude that they, too, were part of the Pythagorean geometry.
Lastly, the Pythagoreans discovered the existence of the incommensurable
or irrational in the particular case of the diagonal of a square in
relation to its side. Aristotle mentions an ancient proof of the
incommensurability of the diagonal with the side by a _reductio ad
absurdum_ showing that, if the diagonal were commensurable with the
side, it would follow that one and the same number is both odd and even.
This proof was doubtless Pythagorean.
A word should be added about the Pythagorean astronomy. Pythagoras was
the first to hold that the earth (and no doubt each of the other
heavenly bodies also) is spherical in shape, and he was aware that the
sun, moon and planets have independent movements of their own in a sense
opposite to that of the daily rotation; but he seems to have kept the
earth in the centre. His successors in the school (one Hicetas of
Syracuse and Philolaus are alternatively credited with this innovation)
actually abandoned the geocentric idea and made the earth, like the sun,
the moon, and the other planets, revolve in a circle round the 'central
fire', in which resided the governing principle ordering and directing
the movement of the universe.
The geometry of which we have so far spoken belongs to the Elements.
But, before the body of the Elements was complete, the Greeks had
advanced beyond the Elements. By the second half of the fifth century B.
C. they had investigated three famous problems in higher geometry, (1)
the squaring of the circle, (2) the trisection of any angle, (3) the
duplication of the cube. The great names belonging to this period are
Hippias of Elis, Hippocrates of Chios, and Democritus.
Hippias of Elis invented a certain curve described by combining two
uniform movements (one angular and the other rectilinear) taking the
same time to complete. Hippias himself used his curve for the
trisection of any angle or the division of it in any ratio; but it was
afterwards employed by Dinostratus, a brother of Eudoxus's pupil
Menaechmus, and by Nicomedes for squaring the circle, whence it got the
name τετραγωνιζουσα {tetragônizousa}, _quadratrix_.
Hippocrates of Chios is mentioned by Aristotle as an instance to prove
that a man may be a distinguished geometer and, at the same time, a fool
in the ordinary affairs of life. He occupies an important place both in
elementary geometry and in relation to two of the higher problems above
mentioned. He was, so far as is known, the first compiler of a book of
Elements; and he was the first to prove the important theorem of Eucl.
XII. 2 that circles are to one another as the squares on their
diameters, from which he further deduced that similar segments of
circles are to one another as the squares on their bases. These
propositions were used by him in his tract on the squaring of _lunes_,
which was intended to lead up to the squaring of the circle. The
essential portions of the tract are preserved in a passage of
Simplicius's commentary on Aristotle's _Physics_, which contains
substantial extracts from Eudemus's lost _History of Geometry_.
Hippocrates showed how to square three particular lunes of different
kinds and then, lastly, he squared the sum of a circle and a certain
lune. Unfortunately the last-mentioned lune was not one of those which
can be squared, so that the attempt to square the circle in this way
failed after all.
Hippocrates also attacked the problem of doubling the cube. There are
two versions of the origin of this famous problem. According to one
story an old tragic poet had represented Minos as having been
dissatisfied with the size of a cubical tomb erected for his son Glaucus
and having told the architect to make it double the size while retaining
the cubical form. The other story says that the Delians, suffering from
a pestilence, consulted the oracle and were told to double a certain
altar as a means of staying the plague. Hippocrates did not indeed solve
the problem of duplication, but reduced it to another, namely that of
finding two mean proportionals in continued proportion between two given
straight lines; and the problem was ever afterwards attacked in this
form. If _x_, _y_ be the two required mean proportionals between two
straight lines _a_, _b_, then _a:x=x:y=y:b_, whence _b/a=(x/a)³_, and,
as a particular case, if _b=2a_, _x³=2a³_, so that, when _x_ is found,
the cube is doubled.
Democritus wrote a large number of mathematical treatises, the titles
only of which are preserved. We gather from one of these titles, 'On
irrational lines and solids', that he wrote on irrationals. Democritus
realized as fully as Zeno, and expressed with no less piquancy, the
difficulty connected with the continuous and the infinitesimal. This
appears from his dilemma about the circular base of a cone and a
parallel section; the section which he means is a section 'indefinitely
near' (as the phrase is) to the base, i. e. the _very next_ section, as
we might say (if there were one). Is it, said Democritus, equal or not
equal to the base? If it is equal, so will the very next section to it
be, and so on, so that the cone will really be, not a cone, but a
cylinder. If it is unequal to the base and in fact less, the surface of
the cone will be jagged, like steps, which is very absurd. We may be
sure that Democritus's work on 'The contact of a circle or a sphere'
discussed a like difficulty.
Lastly, Archimedes tells us that Democritus was the first to state,
though he could not give a rigorous proof, that the volume of a cone or
a pyramid is one-third of that of the cylinder or prism respectively on
the same base and having equal height, theorems first proved by Eudoxus.
We come now to the time of Plato, and here the great names are Archytas,
Theodoras of Cyrene, Theaetetus, and Eudoxus.
Archytas (about 430-360 B. C.) wrote on music and the numerical ratios
corresponding to the intervals of the tetrachord. He is said to have
been the first to write a treatise on mechanics based on mathematical
principles; on the practical side he invented a mechanical dove which
would fly. In geometry he gave the first solution of the problem of the
two mean proportionals, using a wonderful construction in three
dimensions which determined a certain point as the intersection of three
surfaces, (1) a certain cone, (2) a half-cylinder, (3) an anchor-ring or
_tore_ with inner diameter _nil_.
Theodorus, Plato's teacher in mathematics, extended the theory of the
irrational by proving incommensurability in certain particular cases
other than that of the diagonal of a square in relation to its side,
which was already known. He proved that the side of a square containing
3 square feet, or 5 square feet, or any non-square number of square feet
up to 17 is incommensurable with one foot, in other words that √3, √5
... √17 are all incommensurable with 1. Theodorus's proof was evidently
not general; and it was reserved for Theaetetus to comprehend all these
irrationals in one definition, and to prove the property generally as it
is proved in Eucl. X. 9. Much of the content of the rest of Euclid's
Book X (dealing with compound irrationals), as also of Book XIII on the
five regular solids, was due to Theaetetus, who is even said to have
discovered two of those solids (the octahedron and icosahedron).
Plato (427-347 B. C.) was probably not an original mathematician, but he
'caused mathematics in general and geometry in particular to make a
great advance by reason of his enthusiasm for them'. He encouraged the
members of his school to specialize in mathematics and astronomy; e. g.
we are told that in astronomy he set it as a problem to all earnest
students to find 'what are the uniform and ordered movements by the
assumption of which the apparent motions of the planets may be
accounted for'. In Plato's own writings are found certain definitions,
e. g. that of a straight line as 'that of which the middle covers the
ends', and some interesting mathematical illustrations, especially that
in the second geometrical passage in the _Meno_ (86E-87C). To Plato
himself are attributed (1) a formula _(n²-1)²+(2n)²=(n²+1)²_ for finding
two square numbers the sum of which is a square number, (2) the
invention of the method of analysis, which he is said to have explained
to Leodamas of Thasos (_mathematical_ analysis was, however, certainly,
in practice, employed long before). The solution, attributed to Plato,
of the problem of the two mean proportionals by means of a frame
resembling that which a shoemaker uses to measure a foot, can hardly be
his.
Eudoxus (408-355 B. C.), an original genius second to none (unless it be
Archimedes) in the history of our subject, made two discoveries of
supreme importance for the further development of Greek geometry.
(1) As we have seen, the discovery of the incommensurable rendered
inadequate the Pythagorean theory of proportion, which applied to
commensurable magnitudes only. It would no doubt be possible, in most
cases, to replace proofs depending on proportions by others; but this
involved great inconvenience, and a slur was cast on geometry generally.
The trouble was remedied once for all by Eudoxus's discovery of the
great theory of proportion, applicable to commensurable and
incommensurable magnitudes alike, which is expounded in Euclid's Book V.
Well might Barrow say of this theory that 'there is nothing in the whole
body of the elements of a more subtile invention, nothing more solidly
established'. The keystone of the structure is the definition of equal
ratios (Eucl. V, Def. 5); and twenty-three centuries have not abated a
jot from its value, as is plain from the facts that Weierstrass repeats
it word for word as his definition of equal numbers, and it corresponds
almost to the point of coincidence with the modern treatment of
irrationals due to Dedekind.
(2) Eudoxus discovered the method of exhaustion for measuring
curvilinear areas and solids, to which, with the extensions given to it
by Archimedes, Greek geometry owes its greatest triumphs. Antiphon the
Sophist, in connexion with attempts to square the circle, had asserted
that, if we inscribe successive regular polygons in a circle,
continually doubling the number of sides, we shall sometime arrive at a
polygon the sides of which will coincide with the circumference of the
circle. Warned by the unanswerable arguments of Zeno against
infinitesimals, mathematicians substituted for this the statement that,
by continuing the construction, we can inscribe a polygon approaching
equality with the circle _as nearly as we please_. The method of
exhaustion used, for the purpose of proof by _reductio ad absurdum_, the
lemma proved in Eucl. X. 1 (to the effect that, if from any magnitude we
subtract not less than half, and then from the remainder not less than
half, and so on continually, there will sometime be left a magnitude
less than any assigned magnitude of the same kind, however small): and
this again depends on an assumption which is practically contained in
Eucl. V, Def. 4, but is generally known as the Axiom of Archimedes,
stating that, if we have two unequal magnitudes, their difference
(however small) can, if continually added to itself, be made to exceed
any magnitude of the same kind (however great).
The method of exhaustion is seen in operation in Eucl. XII. 1-2, 3-7
Cor., 10, 16-18. Props. 3-7 Cor. and Prop. 10 prove that the volumes of
a pyramid and a cone are one-third of the prism and cylinder
respectively on the same base and of equal height; and Archimedes
expressly says that these facts were first proved by Eudoxus.
In astronomy Eudoxus is famous for the beautiful theory of concentric
spheres which he invented to explain the apparent motions of the
planets and, particularly, their apparent stationary points and
retrogradations. The theory applied also to the sun and moon, for each
of which Eudoxus employed three spheres. He represented the motion of
each planet as produced by the rotations of four spheres concentric with
the earth, one within the other, and connected in the following way.
Each of the inner spheres revolves about a diameter the ends of which
(poles) are fixed on the next sphere enclosing it. The outermost sphere
represents the daily rotation, the second a motion along the zodiac
circle; the poles of the third sphere are fixed on the latter circle;
the poles of the fourth sphere (carrying the planet fixed on its
equator) are so fixed on the third sphere, and the speeds and directions
of rotation so arranged, that the planet describes on the second sphere
a curve called the _hippopede_ (horse-fetter), or a figure of eight,
lying along and longitudinally bisected by the zodiac circle. The whole
arrangement is a marvel of geometrical ingenuity.
Heraclides of Pontus (about 388-315 B. C.), a pupil of Plato, made a
great step forward in astronomy by his declaration that the earth
rotates on its own axis once in 24 hours, and by his discovery that
Mercury and Venus revolve about the sun like satellites.
Menaechmus, a pupil of Eudoxus, was the discoverer of the conic
sections, two of which, the parabola and the hyperbola, he used for
solving the problem of the two mean proportionals. If _a:x=x:y=y:b_,
then _x²=ay_, _y²=bx_ and _xy=ab_. These equations represent, in
Cartesian co-ordinates, and with rectangular axes, the conics by the
intersection of which two and two Menaechmus solved the problem; in the
case of the rectangular hyperbola it was the asymptote-property which he
used.
We pass to Euclid's times. A little older than Euclid, Autolycus of
Pitane wrote two books, _On the Moving Sphere_, a work on Sphaeric for
use in astronomy, and _On Risings and Settings_. The former work is the
earliest Greek textbook which has reached us intact. It was before
Euclid when he wrote his _Phaenomena_, and there are many points of
contact between the two books.
Euclid flourished about 300 B. C. or a little earlier. His great work,
the _Elements_ in thirteen Books, is too well known to need description.
No work presumably, except the Bible, has had such a reign; and future
generations will come back to it again and again as they tire of the
variegated substitutes for it and the confusion resulting from their
bewildering multiplicity. After what has been said above of the growth
of the Elements, we can appreciate the remark of Proclus about Euclid,
'who put together the Elements, collecting many of Eudoxus's theorems,
perfecting many of Theaetetus's and also bringing to irrefragable
demonstration the things which were only somewhat loosely proved by his
predecessors'. Though a large portion of the subject-matter had been
investigated by those predecessors, everything goes to show that the
whole arrangement was Euclid's own; it is certain that he made great
changes in the order of propositions and in the proofs, and that his
innovations began at the very beginning of Book I.
Euclid wrote other books on both elementary and higher geometry, and on
the other mathematical subjects known in his day. The elementary
geometrical works include the _Data_ and _On Divisions_ (_of figures_),
the first of which survives in Greek and the second in Arabic only; also
the _Pseudaria_, now lost, which was a sort of guide to fallacies in
geometrical reasoning. The treatises on higher geometry are all lost;
they include (1) the _Conics_ in four Books, which covered almost the
same ground as the first three Books of Apollonius's _Conics_, although
no doubt, for Euclid, the conics were still, as with his predecessors,
sections of a right-angled, an obtuse-angled, and an acute-angled cone
respectively made by a plane perpendiular to a generator in each case;
(2) the _Porisms_ in three Books, the importance and difficulty of which
can be inferred from Pappus's account of it and the lemmas which he
gives for use with it; (3) the _Surface-Loci_, to which again Pappus
furnishes lemmas; one of these implies that Euclid assumed as known the
focus-directrix property of the three conics, which is absent from
Apollonius's _Conics_.
In applied mathematics Euclid wrote (1) the _Phaenomena_, a work on
spherical astronomy in which ὁ ὁριζων {ho horizôn} (without κυκλος
{kyklos} or any qualifying words) appears for the first time in the
sense of _horizon_; (2) the _Optics_, a kind of elementary treatise on
perspective: these two treatises are extant in Greek; (3) a work on the
Elements of Music. The _Sectio Canonis_, which has come down under the
name of Euclid, can, however, hardly be his in its present form.
In the period between Euclid and Archimedes comes Aristarchus of Samos
(about 310-230 B. C.), famous for having anticipated Copernicus.
Accepting Heraclides's view that the earth rotates about its own axis,
Aristarchus went further and put forward the hypothesis that the sun
itself is at rest, and that the earth, as well as Mercury, Venus, and
the other planets, revolve in circles about the sun. We have this on the
unquestionable authority of Archimedes, who was only some twenty-five
years later, and who must have seen the book containing the hypothesis
in question. We are told too that Cleanthes the Stoic thought that
Aristarchus ought to be indicted on the charge of impiety for setting
the Hearth of the Universe in motion.
One work of Aristarchus, _On the sizes and distances of the Sun and
Moon_, which is extant in Greek, is highly interesting in itself, though
it contains no word of the heliocentric hypothesis. Thoroughly classical
in form and style, it lays down certain hypotheses and then deduces
therefrom, by rigorous geometry, the sizes and distances of the sun and
moon. If the hypotheses had been exact, the results would have been
correct too; but Aristarchus in fact assumed a certain angle to be 87°
which is really 89° 50', and the angle subtended at the centre of the
earth by the diameter of either the sun or the moon to be 2°, whereas we
know from Archimedes that Aristarchus himself discovered that the latter
angle is only 1/2°. The effect of Aristarchus's geometry is to find
arithmetical limits to the values of what are really trigonometrical
ratios of certain small angles, namely
1/18 > sin 3° > 1/20, 1/45 > sin 1° > 1/60, 1 > cos 1° > 89/90.
The main results obtained are (1) that the diameter of the sun is
between 18 and 20 times the diameter of the moon, (2) that the diameter
of the moon is between 2/45ths and 1/30th of the distance of the centre
of the moon from our eye, and (3) that the diameter of the sun is
between 19/3rds and 43/6ths of the diameter of the earth. The book
contains a good deal of arithmetical calculation.
Archimedes was born about 287 B. C. and was killed at the sack of
Syracuse by Marcellus's army in 212 B. C. The stories about him are well
known, how he said 'Give me a place to stand on, and I will move the
earth' (πα βω και κινω ταν γαν {pa bô kai kinô tan gan}); how, having
thought of the solution of the problem of the crown when in the bath, he
ran home naked shouting ἑυρηκα, ἑυρηκα {heurêka, heurêka}; and how, the
capture of Syracuse having found him intent on a figure drawn on the
ground, he said to a Roman soldier who came up, 'Stand away, fellow,
from my diagram.' Of his work few people know more than that he invented
a tubular screw which is still used for pumping water, and that for a
long time he foiled the attacks of the Romans on Syracuse by the
mechanical devices and engines which he used against them. But he
thought meanly of these things, and his real interest was in pure
mathematical speculation; he caused to be engraved on his tomb a
representation of a cylinder circumscribing a sphere, with the ratio
3/2 which the cylinder bears to the sphere: from which we infer that he
regarded this as his greatest discovery.
Archimedes's works are all original, and are perfect models of
mathematical exposition; their wide range will be seen from the list of
those which survive: _On the Sphere and Cylinder_ I, II, _Measurement of
a Circle_, _On Conoids and Spheroids_, _On Spirals_, _On Plane
Equilibriums_ I, II, the _Sandreckoner_, _Quadrature of the Parabola_,
_On Floating Bodies_ I, II, and lastly the _Method_ (only discovered in
1906). The difficult Cattle-Problem is also attributed to him, and a
_Liber Assumptorum_ which has reached us through the Arabic, but which
cannot be his in its present form, although some of the propositions in
it (notably that about the 'Salinon', salt-cellar, and others about
circles inscribed in the αρβηλος {arbêlos}, shoemaker's knife) are quite
likely to be of Archimedean origin. Among lost works were the
_Catoptrica_, _On Sphere-making_, and investigations into polyhedra,
including thirteen semi-regular solids, the discovery of which is
attributed by Pappus to Archimedes.
Speaking generally, the geometrical works are directed to the
measurement of curvilinear areas and volumes; and Archimedes employs a
method which is a development of Eudoxus's method of exhaustion. Eudoxus
apparently approached the figure to be measured from below only, i. e.
by means of figures successively inscribed to it. Archimedes approaches
it from both sides by successively inscribing figures and circumscribing
others also, thereby compressing them, as it were, until they coincide
as nearly as we please with the figure to be measured. In many cases his
procedure is, when the analytical equivalents are set down, seen to
amount to real _integration_; this is so with his investigation of the
areas of a parabolic segment and a spiral, the surface and volume of a
sphere, and the volume of any segments of the conoids and spheroids.
The newly-discovered _Method_ is especially interesting as showing how
Archimedes originally obtained his results; this was by a clever
mechanical method of (theoretically) _weighing_ infinitesimal elements
of the figure to be measured against elements of another figure the area
or content of which (as the case may be) is known; it amounts to an
_avoidance_ of integration. Archimedes, however, would only admit that
the mechanical method is useful for finding results; he did not consider
them proved until they were established geometrically.
In the _Measurement of a Circle_, after proving by exhaustion that the
area of a circle is equal to a right-angled triangle with the
perpendicular sides equal respectively to the radius and the
circumference of the circle, Archimedes finds, by sheer calculation,
upper and lower limits to the ratio of the circumference of a circle to
its diameter (what we call π {p}). This he does by inscribing and
circumscribing regular polygons of 96 sides and calculating
approximately their respective perimeters. He begins by assuming as
known certain approximate values for √3, namely 1351/780 > √3 > 265/153,
and his calculations involve approximating to the square roots of
several large numbers (up to seven digits). The text only gives the
results, but it is evident that the extraction of square roots presented
no difficulty, notwithstanding the comparative inconvenience of the
alphabetic system of numerals. The result obtained is well known, namely
3-1/7 > π {p} > 3-10/71.
The _Plane Equilibriums_ is the first scientific treatise on the first
principles of mechanics, which are established by pure geometry. The
most important result established in Book I is the principle of the
lever. This was known to Plato and Aristotle, but they had no real
proof. The Aristotelian _Mechanics_ merely 'refers' the lever 'to the
circle', asserting that the force which acts at the greater distance
from the fulcrum moves the system more easily because it describes a
greater circle. Archimedes also finds the centre of gravity of a
parallelogram, a triangle, a trapezium and finally (in Book II) of a
parabolic segment and of a portion of it cut off by a straight line
parallel to the base.
The _Sandreckoner_ is remarkable for the development in it of a system
for expressing very large numbers by _orders_ and _periods_ based on
powers of myriad-myriads (10,000²). It also contains the important
reference to the heliocentric theory of the universe put forward by
Aristarchus of Samos in a book of 'hypotheses', as well as historical
details of previous attempts to measure the size of the earth and to
give the sizes and distances of the sun and moon.
Lastly, Archimedes invented the whole science of hydrostatics. Beginning
the treatise _On Floating Bodies_ with an assumption about uniform
pressure in a fluid, he first proves that the surface of a fluid at rest
is a sphere with its centre at the centre of the earth. Other
propositions show that, if a solid floats in a fluid, the weight of the
solid is equal to that of the fluid displaced, and, if a solid heavier
than a fluid is weighed in it, it will be lighter than its true weight
by the weight of the fluid displaced. Then, after a second assumption
that bodies which are forced upwards in a fluid are forced upwards along
the perpendiculars to the surface which pass through their centres of
gravity, Archimedes deals with the position of rest and stability of a
segment of a sphere floating in a fluid with its base entirely above or
entirely below the surface. Book II is an extraordinary _tour de force_,
investigating fully all the positions of rest and stability of a right
segment of a paraboloid floating in a fluid according (1) to the
relation between the axis of the solid and the parameter of the
generating parabola, and (2) to the specific gravity of the solid in
relation to the fluid; the term 'specific gravity' is not used, but the
idea is fully expressed in other words.
Almost contemporary with Archimedes was Eratosthenes of Cyrene, to whom
Archimedes dedicated the _Method_; the preface to this work shows that
Archimedes thought highly of his mathematical ability. He was indeed
recognized by his contemporaries as a man of great distinction in all
branches, though the names Beta and Pentathlos[4] applied to him
indicate that he just fell below the first rank in each subject. Ptolemy
Euergetes appointed him to be tutor to his son (Philopator), and he
became librarian at Alexandria; he recognized his obligation to Ptolemy
by erecting a column with a graceful epigram. In this epigram he
referred to the earlier solutions of the problem of duplicating the cube
or finding the two mean proportionals, and advocated his own in
preference, because it would give any number of means; on the column was
fixed a bronze representation of his appliance, a frame with
right-angled triangles (or rectangles) movable along two parallel
grooves and over one another, together with a condensed proof. The
_Platonicus_ of Eratosthenes evidently dealt with the fundamental
notions of mathematics in connexion with Plato's philosophy, and seems
to have begun with the story of the origin of the duplication problem.
[4] This word primarily means an all-round athlete, a winner in all
five of the sports constituting the πενταθλον {pentathlon}, namely
jumping, discus-throwing, running, wrestling, and boxing (or
javelin-throwing).
The most famous achievement of Eratosthenes was his measurement of the
earth. Archimedes quotes an earlier measurement which made the
circumference of the earth 300,000 stades. Eratosthenes improved upon
this. He observed that at the summer solstice at Syene, at noon, the sun
cast no shadow, while at the same moment the upright gnomon at
Alexandria cast a shadow corresponding to an angle between the gnomon
and the sun's rays of 1/50th of four right angles. The distance between
Syene and Alexandria being known to be 5,000 stades, this gave for the
circumference of the earth 250,000 stades, which Eratosthenes seems
later, for some reason, to have changed to 252,000 stades. On the most
probable assumption as to the length of the stade used, the 252,000
stades give about 7,850 miles, only 50 miles less than the true polar
diameter.
In the work _On the Measurement of the Earth_ Eratosthenes is said to
have discussed other astronomical matters, the distance of the tropic
and polar circles, the sizes and distances of the sun and moon, total
and partial eclipses, &c. Besides other works on astronomy and
chronology, Eratosthenes wrote a _Geographica_ in three books, in which
he first gave a history of geography up to date and then passed on to
mathematical geography, the spherical shape of the earth, &c., &c.
Apollonius of Perga was with justice called by his contemporaries the
'Great Geometer', on the strength of his great treatise, the _Conics_.
He is mentioned as a famous astronomer of the reign of Ptolemy Euergetes
(247-222 B. C.); and he dedicated the fourth and later Books of the
_Conics_ to King Attalus I of Pergamum (241-197 B. C.).
The _Conics_, a colossal work, originally in eight Books, survives as to
the first four Books in Greek and as to three more in Arabic, the eighth
being lost. From Apollonius's prefaces we can judge of the relation of
his work to Euclid's _Conics_, the content of which answered to the
first three Books of Apollonius. Although Euclid knew that an ellipse
could be otherwise produced, e. g. as an oblique section of a right
cylinder, there is no doubt that he produced all three conics from right
cones like his predecessors. Apollonius, however, obtains them in the
most general way by cutting any oblique cone, and his original axes of
reference, a diameter and the tangent at its extremity, are in general
oblique; the fundamental properties are found with reference to these
axes by 'application of areas', the three varieties of which,
_application_ (παραβολη {parabolê}), application with an _excess_
(ὑπερβολη {hyperbolê}) and application with a _deficiency_ (ελλειψις
{elleipsis}), give the properties of the three curves respectively and
account for the names _parabola_, _hyperbola_, and _ellipse_, by which
Apollonius called them for the first time. The principal axes only
appear, as a particular case, after it has been shown that the curves
have a like property when referred to any other diameter and the tangent
at its extremity, instead of those arising out of the original
construction. The first four Books constitute what Apollonius calls an
elementary introduction; the remaining Books are specialized
investigations, the most important being Book V (on normals) and Book
VII (mainly on conjugate diameters). Normals are treated, not in
connexion with tangents, but as _minimum_ or _maximum_ straight lines
drawn to the curves from different points or classes of points.
Apollonius discusses such questions as the number of normals that can be
drawn from one point (according to its position) and the construction of
all such normals. Certain propositions of great difficulty enable us to
deduce quite easily the Cartesian equations to the _evolutes_ of the
three conics.
Several other works of Apollonius are described by Pappus as forming
part of the 'Treasury of Analysis'. All are lost except the _Sectio
Rationis_ in two Books, which survives in Arabic and was published in a
Latin translation by Halley in 1706. It deals with all possible cases of
the general problem 'given two straight lines either parallel or
intersecting, and a fixed point on each, to draw through any given point
a straight line which shall cut off intercepts from the two lines
(measured from the fixed points) bearing a given ratio to one another'.
The lost treatise _Sectio Spatii_ dealt similarly with the like problem
in which the intercepts cut off have to contain a given rectangle.
The other treatises included in Pappus's account are (1) On _Determinate
Section_; (2) _Contacts_ or _Tangencies_, Book II of which is entirely
devoted to the problem of drawing a circle to touch three given circles
(Apollonius's solution can, with the aid of Pappus's auxiliary
propositions, be satisfactorily restored); (3) _Plane Loci_, i. e. loci
which are straight lines or circles; (4) Νευσεις {Neuseis},
_Inclinationes_ (the general problem called a νευσις {neusis} being to
insert between two lines, straight or curved, a straight line of given
length _verging_ to a given point, i. e. so that, if produced, it passes
through the point, Apollonius restricted himself to cases which could be
solved by 'plane' methods, i. e. by the straight line and circle only).
Apollonius is also said to have written (5) a _Comparison of the
dodecahedron with the icosahedron_ (inscribed in the same sphere), in
which he proved that their surfaces are in the same ratio as their
volumes; (6) _On the cochlias_ or cylindrical helix; (7) a 'General
Treatise', which apparently dealt with the fundamental assumptions, &c.,
of elementary geometry; (8) a work on _unordered irrationals_, i. e.
irrationals of more complicated form than those of Eucl. Book X; (9) _On
the burning-mirror_, dealing with spherical mirrors and probably with
mirrors of parabolic section also; (10) ωκυτοκιον {ôkytokion} ('quick
delivery'). In the last-named work Apollonius found an approximation to
π {p} closer than that in Archimedes's _Measurement of a Circle_; and
possibly the book also contained Apollonius's exposition of his notation
for large numbers according to 'tetrads' (successive powers of the
myriad).
In astronomy Apollonius is said to have made special researches
regarding the moon, and to have been called ε {e} (Epsilon) because the
form of that letter is associated with the moon. He was also a master of
the theory of epicycles and eccentrics.
With Archimedes and Apollonius Greek geometry reached its culminating
point; indeed, without some more elastic notation and machinery such as
algebra provides, geometry was practically at the end of its resources.
For some time, however, there were capable geometers who kept up the
tradition, filling in details, devising alternative solutions of
problems, or discovering new curves for use or investigation.
Nicomedes, probably intermediate in date between Eratosthenes and
Apollonius, was the inventor of the _conchoid_ or _cochloid_, of which,
according to Pappus, there were three varieties. Diocles (about the end
of the second century B. C.) is known as the discoverer of the _cissoid_
which was used for duplicating the cube. He also wrote a book περι
πυρειων {peri pyreiôn}, _On burning-mirrors_, which probably discussed,
among other forms of mirror, surfaces of parabolic or elliptic section,
and used the focal properties of the two conics; it was in this work
that Diocles gave an independent and clever solution (by means of an
ellipse and a rectangular hyperbola) of Archimedes's problem of cutting
a sphere into two segments in a given ratio. Dionysodorus gave a
solution by means of conics of the auxiliary cubic equation to which
Archimedes reduced this problem; he also found the solid content of a
_tore_ or anchor-ring.
Perseus is known as the discoverer and investigator of the _spiric
sections_, i. e. certain sections of the σπειρα {speira}, one variety of
which is the _tore_. The _spire_ is generated by the revolution of a
circle about a straight line in its plane, which straight line may
either be external to the circle (in which case the figure produced is
the tore), or may cut or touch the circle.
Zenodorus was the author of a treatise on _Isometric figures_, the
problem in which was to compare the content of different figures, plane
or solid, having equal contours or surfaces respectively.
Hypsicles (second half of second century B. C.) wrote what became known
as 'Book XIV' of the _Elements_ containing supplementary propositions on
the regular solids (partly drawn from Aristaeus and Apollonius); he
seems also to have written on polygonal numbers. A mediocre astronomical
work (Αναφορικος {Anaphorikos}) attributed to him is the first Greek
book in which we find the division of the zodiac circle into 360 parts
or degrees.
Posidonius the Stoic (about 135-51 B. C.) wrote on geography and
astronomy under the titles _On the Ocean_ and περι μετεωρων {peri
meteôrôn}. He made a new but faulty calculation of the circumference of
the earth (240,000 stades). _Per contra_, in a separate tract on the
size of the sun (in refutation of the Epicurean view that it is as big
as it _looks_), he made assumptions (partly guesswork) which give for
the diameter of the sun a figure of 3,000,000 stades (39-1/4 times the
diameter of the earth), a result much nearer the truth than those
obtained by Aristarchus, Hipparchus, and Ptolemy. In elementary geometry
Posidonius gave certain definitions (notably of parallels, based on the
idea of equidistance).
Geminus of Rhodes, a pupil of Posidonius, wrote (about 70 B. C.) an
encyclopaedic work on the classification and content of mathematics,
including the history of each subject, from which Proclus and others
have preserved notable extracts. An-Nairīzī (an Arabian commentator on
Euclid) reproduces an attempt by one 'Aganis', who appears to be
Geminus, to prove the parallel-postulate.
But from this time onwards the study of higher geometry (except
sphaeric) seems to have languished, until that admirable mathematician,
Pappus, arose (towards the end of the third century A. D.) to revive
interest in the subject. From the way in which, in his great
_Collection_, Pappus thinks it necessary to describe in detail the
contents of the classical works belonging to the 'Treasury of Analysis'
we gather that by his time many of them had been lost or forgotten, and
that he aimed at nothing less than re-establishing geometry at its
former level. No one could have been better qualified for the task.
Presumably such interest as Pappus was able to arouse soon flickered
out; but his _Collection_ remains, after the original works of the great
mathematicians, the most comprehensive and valuable of all our sources,
being a handbook or guide to Greek geometry and covering practically the
whole field. Among the original things in Pappus's _Collection_ is an
enunciation which amounts to an anticipation of what is known as
Guldin's Theorem.
It remains to speak of three subjects, trigonometry (represented by
Hipparchus, Menelaus, and Ptolemy), mensuration (in Heron of
Alexandria), and algebra (Diophantus).
Although, in a sense, the beginnings of trigonometry go back to
Archimedes (_Measurement of a Circle_), Hipparchus was the first person
who can be proved to have used trigonometry systematically. Hipparchus,
the greatest astronomer of antiquity, whose observations were made
between 161 and 126 B. C., discovered the precession of the equinoxes,
calculated the mean lunar month at 29 days, 12 hours, 44 minutes, 2-1/2
seconds (which differs by less than a second from the present accepted
figure!), made more correct estimates of the sizes and distances of the
sun and moon, introduced great improvements in the instruments used for
observations, and compiled a catalogue of some 850 stars; he seems to
have been the first to state the position of these stars in terms of
latitude and longitude (in relation to the ecliptic). He wrote a
treatise in twelve Books on Chords in a Circle, equivalent to a table of
trigonometrical sines. For calculating arcs in astronomy from other arcs
given by means of tables he used propositions in spherical trigonometry.
The _Sphaerica_ of Theodosius of Bithynia (written, say, 20 B. C.)
contains no trigonometry. It is otherwise with the _Sphaerica_ of
Menelaus (fl. A. D. 100) extant in Arabic; Book I of this work contains
propositions about spherical triangles corresponding to the main
propositions of Euclid about plane triangles (e.g. congruence theorems
and the proposition that in a spherical triangle the three angles are
together greater than two right angles), while Book III contains genuine
spherical trigonometry, consisting of 'Menelaus's Theorem' with
reference to the sphere and deductions therefrom.
Ptolemy's great work, the _Syntaxis_, written about A. D. 150 and
originally called Μαθηματικη συνταξις {Mathêmatikê syntaxis}, came to be
known as Μεγαλη συνταξις {Megalê syntaxis}; the Arabs made up from the
superlative μεγιστος {megistos} the word al-Majisti which became
_Almagest_.
Book I, containing the necessary preliminaries to the study of the
Ptolemaic system, gives a Table of Chords in a circle subtended by
angles at the centre of ½° increasing by half-degrees to 180°. The
circle is divided into 360 μοιραι {moirai}, parts or degrees, and the
diameter into 120 parts (τμηματα {tmêmata}); the chords are given in
terms of the latter with sexagesimal fractions (e. g. the chord
subtended by an angle of 120° is 103^{p} 53′ 23″). The Table of Chords
is equivalent to a table of the _sines_ of the halves of the angles in
the table, for, if (crd. 2 α {a}) represents the chord subtended by an
angle of 2 α {a} (crd. 2 α {a})/120 = sin α {a}. Ptolemy first gives the
minimum number of geometrical propositions required for the calculation
of the chords. The first of these finds (crd. 36°) and (crd. 72°) from
the geometry of the inscribed pentagon and decagon; the second
('Ptolemy's Theorem' about a quadrilateral in a circle) is equivalent to
the formula for sin (θ-φ) {th-ph}, the third to that for sin ½ θ {th}.
From (crd. 72°) and (crd. 60°) Ptolemy, by using these propositions
successively, deduces (crd. 1½°) and (crd. ¾°), from which he obtains
(crd. 1°) by a clever interpolation. To complete the table he only needs
his fourth proposition, which is equivalent to the formula for cos (θ+φ)
{th+ph}.
Ptolemy wrote other minor astronomical works, most of which survive in
Greek or Arabic, an _Optics_ in five Books (four Books almost complete
were translated into Latin in the twelfth century), and an attempted
proof of the parallel-postulate which is reproduced by Proclus.
Heron of Alexandria (date uncertain; he may have lived as late as the
third century A. D.) was an almost encyclopaedic writer on mathematical
and physical subjects. He aimed at practical utility rather than
theoretical completeness; hence, apart from the interesting collection
of _Definitions_ which has come down under his name, and his commentary
on Euclid which is represented only by extracts in Proclus and
an-Nairīzī, his geometry is mostly mensuration in the shape of numerical
examples worked out. As these could be indefinitely multiplied, there
was a temptation to add to them and to use Heron's name. However much of
the separate works edited by Hultsch (the _Geometrica_, _Geodaesia_,
_Stereometrica_, _Mensurae_, _Liber geëponicus_) is genuine, we must now
regard as more authoritative the genuine _Metrica_ discovered at
Constantinople in 1896 and edited by H. Schöne in 1903 (Teubner). Book I
on the measurement of areas is specially interesting for (1) its
statement of the formula used by Heron for finding approximations to
surds, (2) the elegant geometrical proof of the formula for the area of
a triangle Δ {D} = √{_s (s-a) (s-b) (s-c)}, a formula now known to be
due to Archimedes, (3) an allusion to limits to the value of π {p} found
by Archimedes and more exact than the 3-1/7 and 3-10/71 obtained in the
_Measurement of a Circle_.
Book I of the _Metrica_ calculates the areas of triangles,
quadrilaterals, the regular polygons up to the dodecagon (the areas even
of the heptagon, enneagon, and hendecagon are approximately evaluated),
the circle and a segment of it, the ellipse, a parabolic segment, and
the surfaces of a cylinder, a right cone, a sphere and a segment
thereof. Book II deals with the measurement of solids, the cylinder,
prisms, pyramids and cones and frusta thereof, the sphere and a segment
of it, the anchor-ring or tore, the five regular solids, and finally the
two special solids of Archimedes's _Method_; full use is made of all
Archimedes's results. Book III is on the division of figures. The plane
portion is much on the lines of Euclid's _Divisions_ (of figures). The
solids divided in given ratios are the sphere, the pyramid, the cone and
a frustum thereof. Incidentally Heron shows how he obtained an
approximation to the cube root of a non-cube number (100). Quadratic
equations are solved by Heron by a regular rule not unlike our method,
and the _Geometrica_ contains two interesting indeterminate problems.
Heron also wrote _Pneumatica_ (where the reader will find such things as
siphons, Heron's Fountain, penny-in-the-slot machines, a fire-engine, a
water-organ, and many arrangements employing the force of steam),
_Automaton-making_, _Belopoeïca_ (on engines of war), _Catoptrica_, and
_Mechanics_. The _Mechanics_ has been edited from the Arabic; it is
(except for considerable fragments) lost in Greek. It deals with the
puzzle of 'Aristotle's Wheel', the parallelogram of velocities,
definitions of, and problems on, the centre of gravity, the distribution
of weights between several supports, the five mechanical powers,
mechanics in daily life (queries and answers). Pappus covers much the
same ground in Book VIII of his _Collection_.
We come, lastly, to Algebra. Problems involving simple equations are
found in the Papyrus Rhind, in the _Epanthema_ of Thymaridas already
referred to, and in the arithmetical epigrams in the Greek Anthology
(Plato alludes to this class of problem in the _Laws_, 819 B, C); the
Anthology even includes two cases of indeterminate equations of the
first degree. The Pythagoreans gave general solutions in rational
numbers of the equations _x²+y²=z²_ and _2x²-y²=±1_, which are
indeterminate equations of the second degree.
The first to make systematic use of symbols in algebraical work was
Diophantus of Alexandria (fl. about A. D. 250). He used (1) a sign for
the unknown quantity, which he calls αριθμος {arithmos}, and compendia
for its powers up to the sixth; (2) a sign ([Transcriber's Note:
Symbol]) with the effect of our _minus_. The latter sign probably
represents ΛΙ {LI}, an abbreviation for the root of the word λειπειν
{leipein} (to be wanting); the sign for αριθμος {arithmos}
([Transcriber's Note: Symbol]) is most likely an abbreviation for the
letters αρ {ar}; the compendia for the powers of the unknown are Δ^Υ
{D^Y} for δυναμις {dynamis}, the square, Κ^Υ {K^Y} for κυβος {kybos},
the cube, and so on. Diophantus shows that he solved quadratic equations
by rule, like Heron. His _Arithmetica_, of which six books only (out of
thirteen) survive, contains a certain number of problems leading to
simple equations, but is mostly devoted to indeterminate or
semi-determinate analysis, mainly of the second degree. The collection
is extraordinarily varied, and the devices resorted to are highly
ingenious. The problems solved are such as the following (fractional as
well as integral solutions being admitted): 'Given a number, to find
three others such that the sum of the three, or of any pair of them,
together with the given number is a square', 'To find four numbers such
that the square of the sum _plus_ or _minus_ any one of the numbers is a
square', 'To find three numbers such that the product of any two _plus_
or _minus_ the sum of the three is a square'. Diophantus assumes as
known certain theorems about numbers which are the sums of two and three
squares respectively, and other propositions in the Theory of Numbers.
He also wrote a book _On Polygonal Numbers_ of which only a fragment
survives.
With Pappus and Diophantus the list of original writers on mathematics
comes to an end. After them came the commentators whose names only can
be mentioned here. Theon of Alexandria, the editor of Euclid, lived
towards the end of the fourth century A. D. To the fifth and sixth
centuries belong Proclus, Simplicius, and Eutocius, to whom we can never
be grateful enough for the precious fragments which they have preserved
from works now lost, and particularly the _History of Geometry_ and the
_History of Astronomy_ by Aristotle's pupil Eudemus.
Such is the story of Greek mathematical science. If anything could
enhance the marvel of it, it would be the consideration of the shortness
of the time (about 350 years) within which the Greeks, starting from the
very beginning, brought geometry to the point of performing operations
equivalent to the integral calculus and, in the realm of astronomy,
actually anticipated Copernicus.