Hypotheses, Postulates, Axioms & Propositional Form¶
Epistemology of First Principles, Compass & Straightedge Postulates, Critique of Postulate 5, Common Notions, and the Six Parts of a Proposition — Friedlein (1873) pp. 178–207 · Book III (Petitions, Axioms & Prop. I Prefatory Sections)
Note
Edition Track (English): Public-domain translation by Thomas Taylor (London, 1788–1789 / 1792), collated and synchronized with Gottfried Friedlein's 1873 Teubner Greek edition (pp. 178–207). Switch tracks above for the Greek (1873) text, Summary, or Notes.
Distinction of Hypotheses, Postulates & Axioms¶
Friedlein pp. 178–184
Concerning Petitions and Axioms.
Since the principles of geometry are triply divided into Hypotheses, Petitions, and Axioms, the difference between these we have explained in the preceding books. But we now intend to discourse more accurately of petition and axiom, as especially necessary to our present design. For hypotheses, which are also called definitions, we have already explained. It is common, therefore, as well to axioms as to petitions, to require no demonstration, and no geometrical faith: but to be received as manifest, and to become the principles of the rest. But they differ mutually from each other, in the same manner in which we have distinguished theorems from problems. For as in theorems we propose to perceive and know that which follows a subject; but in problems we are ordered to compare and do something: in the same manner also in axioms, we must receive whatever is manifest of itself, and easily apprehended by our untaught conceptions; but in petitions we must receive whatever is easy to be done and compared, (since in admitting these, thought is not fatigued) and whatever requires no variety, and no kind of construction. Hence evident and indemonstrable cognition, and unconstructed assumption, distinguish petitions from axioms. Just as demonstrative cognition, and an assumption of things sought, together with construction, separates theorems from problems. For it is every where requisite, that principles in simplicity, indemonstrability, and self-evidence, should excel things posterior to principles. For universally (says Speusippus) of the things which cogitation pursues, some of its energies it produces without a various progression, prepares them for future enquiry, and has a more evident apprehension of these than of visible objects: but others which it is not able immediately to follow, by a transition proceeding from their nature, these it endeavours by consequence to pursue. Thus for example, to draw a right line from one point to another, it receives as evident, and easy to be done. For since in this case the line is composed from the indeclinable flux of a point, and at the same time advances in an orderly progression, because it no where more or less declines, it necessarily falls in another point. Again, if one extremity of a right line abiding, the other is moved about it, it will describe a circle without any labour. But if any one wishes to describe a helix of one revolution, it requires a more various operation. For it is generated by various motions. Likewise if any one wishes to construct an equilateral triangle, he will require a certain method for its construction. For the geometrical intellect says, when I understand a right line, which abides according to one of its extremities, but is moved about it according to the other, and at the same time conceive a point, which is moved in the line from the abiding extreme, I have described a helix of one revolution. For when at the same time both the extremity of the right line, which describes the circle, and the point which is moved in the right line, arrive at the same point, and coincide, they produce for me such a helix. And again, when I describe equal circles, and draw right lines from the common section to the centre of the circles, and a right line from one centre to the other, I shall have an equilateral triangle. The production of these, therefore, is very remote from a simple apprehension, and primary notion. For we are content to pursue the progressions of their origin. Hence it happens that these are compared with greater ease or difficulty, and are exhibited with many or fewer mediums, according to the habit of those who enter on this undertaking: but that they require demonstration and construction, on account of the property of the things sought, which wants the evidence of petitions and axioms.
Petition, therefore, and axiom, are simple and easy to be apprehended. But petition, indeed, commands us to fabricate, and provide a certain matter, in order to the assignation of the symptom, which possesses an easy and simple apprehension: but axiom pronounces a certain essential accident, of itself known to the hearers. As that fire is hot, or any other of those manifest truths, he who doubts of which, we consider as either wanting sense or punishment. Hence, petition and axiom are of the same genus; but they differ in the above-mentioned manner. For each is an indemonstrable principle, but this after one mode, and that after another, as we have already observed. But some think that all these should be called petitions, in the same manner as all problems, things sought. For Archimedes beginning his book of Equiponderants, we desire it may be granted (says he) that things equally heavy, from equal lengths, will equally ponderate; though some would rather chuse to call this an axiom. But others call all these axioms, in the same manner as they denominate every thing a theorem, which requires demonstration. For, according to the same proportion, as it seems they pass from proper names to such as are common. Nevertheless, as a problem differs from a theorem, so petition from axiom: though both these last are indemonstrable, and the former require demonstration. And the one, indeed, is assumed as easy to be done, but the other is granted as easy to be known by the common consent of all men. After this manner, therefore, Geminus distinguishes petitions from axioms.
But others will perhaps say, that petitions are indeed proper to the geometrical matter: but that axioms are common to the universal theory, which is conversant about the how-much, and the how-many. For the geometrician knows that which requires that all right angles are equal, and that every finite right line may be produced straight forwards: but that which says, things equal to one and the same are equal to each other, is a common conception, which not only the arithmetician employs, but every one endued with science, accommodating that which is common to his own particular matter. But Aristotle (as we have before observed) says, that petition, since it is demonstrable, is not granted by the hearer, yet is received as a principle: but that axiom is of itself indemonstrable, and that this is confessed by all, according to habit, though some, for the sake of disputation, have doubted its evidence. Since then, there are these three differences, according to the first, which by operating, and knowledge only distinguishes petition from axiom, it is manifest that which says all right angles are mutually equal, is not a petition. Nor the fifth, which says, if a right line falling on two right lines makes the internal angles towards the same parts less than two right, those right lines infinitely produced, shall coincide towards the parts in which the angles less than two right subsist. For these are neither assumed in construction, nor do they command any thing to be done: but they exhibit a certain symptom, inherent in right angles, and in right lines, departing from angles less than two right. But, according to the second difference, that will not be an axiom which says, that two right lines cannot comprehend space, which some at present consider as an axiom. For this is proper to the geometric matter, as likewise that which affirms that all right angles are equal. But according to the third difference, which is Aristotelic, all those which produce their own credibility by a certain demonstration, are petitions; but whatever are indemonstrable, are axioms. Apollonius, therefore, in vain endeavours to deliver the demonstrations of axioms: for Geminus very properly observes, that some have attempted demonstrations of indemonstrables, and have endeavoured from more unknown mediums, to prove things manifest to all, into which error Apollonius has fallen, who wishes to prove the axiom true, which says, that things equal to one, and the same, are equal to each other: but that others assume in the place of indemonstrables, things requiring demonstration. As is the case with Euclid himself, in the fourth and fifth petition. For some say, that this last, as ambiguous, requires demonstration. Indeed, is it not ridiculous, that theorems should be assigned as indemonstrable, the converse of which are demonstrable? For that the internal angles of coincident right lines are less than two right, Euclid himself shews in the theorem, which says, that two angles of every triangle, however taken, are less than two right: besides, it may be perspicuously shewn, that not every thing equal to a right angle is a right angle. Hence, says Geminus, the converse of these are not to be granted indemonstrable. It seems therefore, according to the ordination of this man, that there are, indeed, three petitions: but that the other two, and the converse of these, require demonstrating science: and that in the axioms, the one which says, that two right lines cannot comprehend space, is superfluously added, since its credibility must be derived from demonstration. And thus much concerning the difference of petitions and axioms. Again, of axioms, some are proper to arithmetic, but others to geometry; and others are common to both: for that which says, every number is measured by unity, is an arithmetical axiom. But that which says equal right lines agree amongst themselves, as also this which affirms that every magnitude is divisible in infinitum, are geometrical axioms: but the one which says that things equal to the same, are mutually equal, and all of this kind are common to both. However, it must be observed, that each science uses such as the last, according to its proper subject; as geometry in magnitudes, but arithmetic in numbers. In like manner of petitions, some are peculiar to particular sciences, but others are common to all. For you must call the petition which requires to be granted, that a number may be divided into the least parts, peculiar to arithmetic: but this, that every finite straight line may be produced straight forwards, peculiar to geometry; and the one which desires us to grant, that quantity may be infinitely increased, common to both; for this passion is equally found to reside in number and magnitude.
Postulates 1–3: Straightedge & Compass Constructibility (Geminus)¶
Euclidean Reference: [I.post.1] · Friedlein pp. 185–187
Editorial Key (Friedlein 185): Quoting Geminus verbatim—«ἑπόμενόν ἐστι τῷ ῥύσιν εἶναι τοῦ σημείου τὴν γραμμὴν καὶ τὴν εὐθεῖαν ὁμαλὴν καὶ ἀπαρέγκλιτον ῥύσιν» (‘it follows from the line’s being the flux of a point, and the straight line a uniform and undeviating flux’)—a passage quoted in Greek by Hoenen (Caput IV, p. 121).
PETITIONS or POSTULATES.
Let it be granted that a straight line may be drawn from any one point to any other point.
That a terminated straight line may be produced to any length in a straight line.
And that a circle may be described from any centre, at any distance from that centre.
According to the opinion of Geminus, these three are necessarily placed among petitions, as well on account of their facility, as because they command us to do something. For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow. If then we conceive a point to be moved with an uninclined, and the shortest motion, we shall fall upon another point, and the first petition will be produced, and we shall understand nothing various or difficult. But if when the right line itself is terminated by a point, we conceive its extremity moved with the shortest indeclinable motion, the second petition will arise from an easy and simple apprehension. But if we again imagine that the terminated right line abides according to its other extreme, but that it moves about that which abides according to the rest, the third petition will be produced; for the centre is the point which abides, but the interval the right line. Since the distance of the centre, from all parts of the circumference, is always equal to the quantity of this interval. But if any one should doubt how we apply motion in geometrical concerns, which have an immoveable existence; and how we can move impartibles, (since this is impossible) we request him to call to mind what we have demonstrated in the beginning of these Commentaries. I mean that the reasons of things subsisting in the phantasy, describe there all the images of cogitation, of which cogitation itself possesses the reason: for an intellect of this kind is an unwritten, ultimate, and passive tablet. Hence, it receives forms from another, accompanied with motion; but we must not understand a corporeal but imaginative motion, and must by no means admit that impartibles are moved with corporeal motions, but that they suffer imaginative progressions. For intellect, though impartible, is moved, yet not according to place, and the phantasy has a proper motion according to the impartible which it contains: but we only regarding corporeal motions, neglect those which are made in things destitute of interval. Impartibles, therefore, are pure from corporeal place, and external motions: but another species of motion, and another place congenial to such motions, is considered in their progressions. For, indeed, we should say, that a point also has position in the phantasy, and should not enquire how an impartible can abide, which is at the same time moved elsewhere, and comprehended by place. Since the place of things, with dimension, possesses itself dimension; but the place of impartibles is destitute of all dimension. The proper species therefore of geometrical concerns, are different from the things they produce; and the motion of bodies is different from that of the forms in the phantasy; and the place of partible is different from that of impartible natures; and it is requisite, by distinguishing these, neither to confound nor disturb the essences of things. But it appears that the first of these three petitions declares to us in images, how the things which are, are contained in their own impartible causes, and are terminated by their immaterial bound; and that previous to their constitution, they are on all sides comprehended in their indivisible embrace: for the points existing, a right line is drawn from the one to the other, is terminated by, and received between them. But the second indicates how the things which are by possessing proper causes proceed to all things, preserving in them a continuation not derived from the natures into which they proceed; but that through a cause of infinite power, they endeavour to permeate every where, with a never-failing progression. And the third petition shadows forth the manner in which these progressions return again to their proper principles: for the convolution of a point producing a circle, by moving about an abiding point, imitates a circular regression. But it is requisite to know, that every line cannot be infinitely produced, for the circle and cissoid, and all such as describe figure, are incapable of this property; as likewise some which produce no figure. For the helix of one revolution cannot be infinitely produced, since it is constituted between two points; nor any other lines similarly formed. But neither is it possible to extend every line from every point, to every point; for every line cannot subsist between all points: and thus much for the three first petitions; let us now proceed to the rest.
Postulate 4: Equality of All Right Angles¶
Euclidean Reference: [I.post.4] · Friedlein pp. 188–191
All right angles are equal to each other.
If the present petition is considered by us as manifest, and as requiring no demonstration, it is not a petition according to the opinion of Geminus, but an axiom; for it affirms a certain essential accident of right angles, not commanding us to perform any thing according to a simple conception. But neither is it a petition according to the division of Aristotle: for petition, according to his opinion, requires some demonstration. But if we should say it is demonstrable, and enquire after its demonstration, yet according to the opinion of Geminus, it ought not to be placed among petitions. The equality, therefore, of right angles, appears from our common conceptions; for since a right angle has the relation of unity or bound to the infinite increase and decrease of the angles on each side, it is equal with respect to every right angle, since we constitute the first right angle after this manner, by a right line making angles on each side of the right line on which it stands equal to each other; but if it be requisite to produce a linear demonstration of this, let there be two right angles, one a b c, the other d e f.
[Illustration]
I say that they are equal; for if they are not equal, one of them must be greater, suppose the angle at b. If then the line d e be adapted to the line a b, the line e f shall fall within. Let it fall as b g, and let the line b c be produced to h; because, then a b c is a right angle, a b h also shall be a right angle, and they shall be mutually equal to each other, from the tenth Definition: the angle a b h therefore, is greater than the angle a b g. Let again the line g b be produced to k, because, therefore a b g is a right angle, the successive angle a b k shall be a right one, and consequently equal to a b g. Hence, the angle a b h, shall be less than the angle a b g; but it was also greater, which is impossible: but this has been shewn by other expositors, and requires no great consideration. But Pappus very properly admonishes us, that the converse of this Petition is not true; I mean, that every thing equal to a right angle, is a right angle; though if it be rectilinear, it is without doubt a right angle. But a curvilinear angle may also be exhibited equal to one that is right: for let there be conceived two equal right lines, a b, and b c, making the angle at the point b, right;
[Illustration]
and on them let the semicircles a e b, b f c, with a proper centre and interval be described; because, therefore, the semicircles are equal, they shall have a mutual congruence, and the angle e b a, is equal to the angle f b c, and a b f is common: the whole right angle, therefore, is equal to the lunular, i.e. to e b f, and yet the lunular is not a right angle. In the same manner, if the angle a b c should be obtuse or acute, a lunular angle may be shewn equal to it (for this is that genus of curvilinear angles which agrees with such as are rectilinear), only this is to be observed, that in a right and obtuse angle, it is requisite to add the middle angle, which is contained by the line a b, and the circumference b f; but in an acute angle to take this away: for the right line c b, in these cases, cuts the circumference b e. The truth of which, will be evident from the following figures:
[Illustration]
[Illustration]
And hence, it appears, that all right angles are mutually equal to each other, and that not every thing equal to a right angle, is consequently a right angle: for if it be not rectilinear, how can it be called right. But it is also manifest from this Petition, that angular rectitude is allied to equality, in the same manner as acuteness and obtuseness are related to inequality. For rectitude and equality, as also similitude, are of the same co-ordination, (for each exists under bound): but acuteness and obtuseness, as also dissimilitude, are of the same series with inequality. For they are all produced from bound and infinite. Hence some, regarding the quantity of angles, say, that a right angle is equal to a right: but others, considering their quality, affirm that one is similar to another. For similitude in qualities is the same as equality in quantities.
Postulate 5: The Parallel Postulate & Asymptotic Curves¶
Euclidean Reference: [I.post.5] · Friedlein pp. 191–193
Editorial Key (Friedlein 191–193): Proclus insists that Postulate 5 must be expunged from the postulates and proved as a theorem, warning via Geminus that asymptotic curves (hyperbola and conchoid of Nicomedes) converge forever without intersecting.
If a right line falling upon two right lines, makes the internal angles towards the same part less than two right, those right lines, if infinitely produced, shall coincide in that part, in which the angles less than two right, are placed.
This ought to be entirely blotted out from the number of Petitions, for it is a theorem including many doubts, which Ptolemy in one of his books proposes to solve; but it requires in its demonstration both many definitions and theorems; and Euclid also exhibits its converse as a theorem. But perhaps some, from an erroneous conception, may think that this should be placed among the petitions, as that which produces credibility of itself, respecting the inclination of right lines, on account of the diminution of two right angles. To such as these, Geminus rightly answers, that from the authors of this science, we learn not entirely to give credit to imaginative probabilities, for the purpose of accomplishing geometrical reasons: for it is similar (says Aristotle) to require demonstrations from a rhetorician, and patiently listen to a geometrician, disputing from probability. And Simmeas in the Phædo of Plato, says, “I know that those who demonstrate from appearances, are vain.” Hence, in the present instance, it is true and necessary that right lines should incline, while right angles are diminished: but this, that the inclining lines, while they are more produced, should at length coincide, is probable, but not necessary, unless some reason demonstrates that this is true in right lines: for there are certain lines infinitely inclining, and never coinciding, and though this appears incredible and admirable, yet it is true, and has been observed in other forms of a line. Is it therefore possible that this can be accomplished in right lines which takes place in others? For before we procure conviction of this, from demonstration, the properties exhibited in other lines molest the phantasy by the contrary images they produce. But if the reasons doubting against the coincidence of lines are very strong, ought we not much more to expel this improbable and irrational supposition from our doctrine? And thus it appears that a demonstration is to be sought for of the present theorem, and that it is foreign from the property of Petitions: but how it is to be demonstrated, and by what reasons the objections urged against it are to be removed, we shall shew in our comment on the proposition, where it is used by the institutor of the elements as manifest. For then it will be necessary to exhibit its evidence, since it does not present itself to our view with indemonstrable clearness, but becomes manifest through the medium of demonstration alone.
Common Notions (Axioms 1–5): Apollonius, Heron & Pappus¶
Euclidean Reference: [I.cn.1] · Friedlein pp. 193–198
AXIOMS.
Things which are equal to the same, are equal to one another.
If equals be added to equals, the wholes are equal.
If equals be taken from equals, the remainders are equal.
If equals are added to unequals, the wholes are unequal.
If equals be taken from unequals, the remainders are unequal.
Things which are double of the same, are equal to one another.
Things which are halves of the same, are equal to one another.
Things which coincide with each other, are mutually equal.
The whole is greater than its part.
Two right lines cannot comprehend space.
These are the things which, according to the opinion of all men, are called indemonstrable axioms, so far as their certainty is admitted by all, and no one disputes their evidence. For propositions also are often simply called axioms, of whatever kind they may be, whether they are immediately proper, or require some declaration; and the Stoics, indeed, are accustomed to call every simple enunciative speech an axiom: and when they write on dialectic arts, they say that they discourse on axioms. But some, distinguishing more accurately axioms from other propositions, give this appellation to a proposition immediate, and producing credibility of itself, on account of its evidence: as also Aristotle and geometricians themselves affirm. For, according to the opinion of these, an axiom is the same as a common conception. By no means, therefore, must we praise Apollonius the geometrician, who writ (as it appears) demonstrations of axioms, because he performs the very opposite to Euclid: for he, indeed, enumerates that which is demonstrable among Petitions; but Apollonius endeavours to find out demonstrations of indemonstrables. But these naturally differ from each other, and the genus of the sciences is different: I mean of the things which take place about immediate propositions, which are entirely subject to our knowledge, on account of their evidence; and of things which use demonstrations, which receive principles from them; and which, when received, they orderly employ in their proper conclusions. But that the demonstration of the first axiom, which Apollonius persuades himself he has invented, possesses a medium, not more known, but more dubious than the conclusion may be known by any one from a slight inspection. For let (says he) a be equal to b, and b to c, I say that a also is equal to c.
[Illustration]
For since a is equal to b, it occupies the same place as b. And because b is equal to c, it occupies the same place as c; and so a occupies the same place as c, they are therefore equal. Now in this demonstration it is requisite that two things must be previously assumed; one, that things occupying the same place, are mutually equal; but the other, that things occupying the same place, with the same thing, mutually occupy the same place: but these are evidently more obscure than the present axiom. For it is proper to enquire how are things, which fill the same place equal, according to the whole, or according to a part; or according to a figure of speech: hence we must by no means admit a transition to place, which is more unknown than the natures it contains; for the invention of its essence is difficult and ambiguous. That we may avoid prolixity, therefore, all axioms are to be delivered as things immediate and self-manifest, since they are of themselves known and credible; for he who brings demonstration to things the most manifest, does not confirm their truth, but diminishes the evidence we possess in the untaught and innate conceptions of the soul: but this is to be received concerning axioms, as a judgment of their peculiarity; and that all of them are of the common kind of the mathematical sciences; and that each of them is said to be verified, not only in magnitudes, but also in numbers, and motions, and times: and this indeed is necessary. For equal and unequal, the whole and part, and the more and the less, are common to discreet and continued quantities. The contemplation, therefore, which is conversant with times and motions, numbers and magnitudes, requires all these, as things evident by their own intrinsic light; and in all of them both that is true, which says, things equal to the same, are equal to one another; as likewise each of the axioms we have assumed: but as they exist in common, each science uses them according to its proper matter, and one indeed, as in magnitudes; but another, as in numbers; and another, as in times; and after this manner in each science, the conclusions become peculiar and apposite, though the axioms are common. Besides, it is likewise requisite not to contract the number of these to the least, as is done by Heron, who only establishes three axioms; for this also is an axiom, the whole is greater than its part, and the geometrician every where assumes this in his demonstrations; as also, that things which mutually coincide, are equal; for this is employed with advantage in the solution of the fourth Proposition. Nor is it proper to join some with others, of which some are proper to the geometric matter, as that two right lines cannot comprehend space, (since axioms are, as we have said, of a common kind); but others are consequent to things established, as that which says, things double of the same, are equal. For this is consequent to the axiom, affirming, that if to equals you add equals, the wholes are equal, since things equal to the half, because they assume the half, become double to the same, and mutually equal, on account of an equal addition: and according to this reason, not only the doubles, but also the triples, and all multiples of the same quantity will appear equal. But with these axioms, Pappus says, that certain others are to be classed, as if unequals are added to equals, the excess of the whole, will be equal to the excess of the adjuncts. And on the contrary, if equals are added to unequals, the excess of the wholes is equal to the excess or difference of the unequals themselves. And these also are manifest from themselves, yet they may be made manifest as follows. Let a be equal to b, and add to each the unequals c d, but let c be greater than d by e, and the remainder be f; because, therefore, a is equal to b, and also f to d; a f will be equal to b d. For if equals are added to equals, the wholes are equals: a c, therefore, exceeds b d, by e only, by which alone c exceeds d. Again, c and d are unequals, to which, let the equals a and b be added, and let e be the excess of c, above d, and the remainder be f; because, therefore, a is equal to b, and f to d a f will be equal to b d; the whole, therefore, a c, will exceed b d, by e only, by which c also exceeds d. These, therefore, are consequent to the aforesaid axioms, and are, not undeservedly, in many copies, omitted. But whatever others he adds to these, have been previously assumed by definitions, to which they are consequent. As for example: that all the parts of a plane and a right line mutually agree; for things placed in their extremities, possess a nature of this kind; and that a point divides a line, but a line a superficies, and a superficies a solid. For all things are divided by the natures by which they are proximately bounded; and that infinite subsists in magnitudes, by addition and diminution, but according to capacity only, in both these respects: for every thing continuous may be infinitely divided and increased. But, as we have summarily spoken
Problems vs. Theorems & The Six Canonical Parts of a Proposition¶
Friedlein pp. 199–207
Editorial Key (Friedlein 203–207 — The Six Canonical Parts): Every complete proposition comprises six formal moments: (1) πρότασις (enunciation, Taylor: proposition), (2) ἔκθεσις (setting-out, Taylor: exposition), (3) διορισμός (specification, Taylor: determination), (4) κατασκευή (construction), (5) ἀπόδειξις (demonstration), and (6) συμπέρασμα (conclusion, Taylor: epilogue).
concerning these, it remains that we consider things consequent to principles; for thus far principles extend themselves. But of those who oppose geometry, some very much doubt concerning principles, endeavouring to shew that the terms have no subsistence, whose arguments, indeed, are known in common, who endeavour to take away all science, and, like hostile foes from a foreign region, demolish the fruits and fecundity of philosophy, as is the case with the Pyrrhonian philosophers; but others only propose to themselves the subversion of geometrical principles, as the Epicureans. Others, again, admitting the principles, affirm, that things consequent to the principles cannot be demonstrated, unless something else is granted, which was not previously assumed in the principles. Zeno of Sidon exercised this mode of contradiction, who was a Sidonian by birth, but of the Epicurean sect, against whom Possidonius wrote an entire book, exhibiting the whole of his imbecile opinion; and thus much may suffice for the difference of opinions concerning principles. We shall shortly consider the troublesome objection of Zeno: but now, after we have briefly resumed the consideration of theorems and problems, their difference, and the divisions they receive, we shall proceed, to an exposition of the things exhibited by the institutor of the elements, gathering the more beautiful observations upon the propositions found in the writings of the antients, and contracting the infinite prolixity of their discourses; but delivering such things as are more artificial, and full of methods producing science, dwelling more on an accurate treatise of things than on the variety of cases and assumptions, to which young men, for the most part, eagerly incline.
[Illustration]
PROPOSITION I. PROBLEM.
Upon a given terminated right line to describe an equilateral triangle.
Since all science is two-fold, and one is conversant about immediate propositions, but another about things, which are exhibited and provided from the propositions, and universally about the consequents to principles; this, again, divides itself in geometrical discourses, into the solution of problems, and the invention of theorems. And problems, indeed, geometry denominates things in which it proposes to procure, manifest, and fabricate that, which, in a certain respect, has no existence; but it calls theorems, things in which it appoints to perceive, know, and demonstrate that which either exists, or does not exist. For problems command us to undertake the origin, positions, applications, descriptions, inscriptions, circumscriptions, coaptations, and contacts of figures, and every thing of this kind: but theorems endeavour to procure our assent to symptoms, and things essentially inherent in the subjects of geometry, and to convince by demonstrations. For geometry discourses concerning every object of enquiry, which is possible to be effected, referring some things to problems, but others to theorems; since it enquires concerning the what, in a two-fold respect: for it either seeks for the reason and intelligence of the thing; or for intelligence, and the essence of the subject. I say, for example, as when it requires what a line of similar parts may be: for in an enquiry of this kind, it either desires to find the definition of such a line, as, that a line of similar parts is that which has all its parts agreeing with all; or to receive the species of lines of similar parts, as that it is either right, or circular, or a cylindric helix. Besides, prior to this, it enquires, by itself, concerning the if, and this especially in its determinations, agitating, whether the object of its enquiry is possible or impossible, what place it possesses, and in how many ways. It likewise seeks concerning the what kind; for when it considers the essential accidents of a triangle, circle, and parallels, it is manifest, that in such cases it seeks after the what kind; but many have thought that geometry very little contemplated the cause, and the why. And of this opinion is Amphinomus, led by the decisions of Aristotle: but (says Geminus) an enquiry into these may be found in geometry. For does it not belong to geometry to enquire for what cause infinite equilateral multangles may be inscribed in circles, but to describe solid equilateral and equiangular multangles, and constructed from similar planes, in spheres, is impossible? To whom does an investigation of this kind belong, except to a geometrician? When, therefore, to geometricians the syllogism is by an impossibility, they alone desire to find the symptom; but when by a principal demonstration, then again if the demonstrations are in that which is particular or partial, the cause is not yet manifest; but if in that which is universal, and in all similars, the why becomes immediately manifest: and thus much concerning objects of enquiry.
But every problem and theorem which receives its completion from its own perfect parts, ought to possess in itself all the following parts: proposition, exposition, determination, construction, demonstration, and conclusion. But of these, proposition informs us what the object of enquiry is from a given datum; for a perfect proposition is composed from both; but exposition receiving the datum essentially, prepares for the question. Again, determination separately explains the thing sought for according to the what; but construction adds to the datum what is wanting to the investigation of the thing sought; and demonstration skilfully collects the proposition from the concessions. But the epilogue, or conclusion, is again converted to the proposition, by confirming that which is exhibited. And so many, indeed, are all the parts of problems and theorems; but proposition, demonstration, and conclusion, are especially necessary, and exist in all; for it is requisite that the thing sought for should be previously known; and that this should be shewn by proper mediums, and that what is exhibited should be concluded; and it is not possible that any one of these three can be wanting; but the rest are, indeed, received in many places; but in many, because they produce no utility, are omitted. For determination and exposition are not found in the problem, which says, to construct an isosceles triangle, which will have each of the angles at the base double of the other; but construction has frequently no subsistence in many theorems, the demonstration being sufficient to exhibit the thing proposed from the data, without any addition. When, therefore, shall we say that exposition fails, when no datum is given in a proposition? Because, though proposition, for the most part, is divided into datum, and the thing sought for, yet this is not always the case; but sometimes the thing sought for, alone affirms that which it is requisite to know or effect, as in the aforesaid problem; for it does not previously say from what datum it is requisite to construct an isosceles triangle, which shall have each of the angles at the base, double of the remaining one; but that it is required to effect this. And here, indeed, the admission of the proposition takes place from things previously known; for we must know the meaning of the terms isosceles, equal and double (since this, as Aristotle observes, is the property of all ratiocinative discipline), yet nothing is subjected to us as in other problems, as in that which says, to bisect a given terminated right line. For here the right line is given, but we are ordered to divide it into two parts; and the datum is separately determined from the object of enquiry. When, therefore, a proposition has both of these, then also determination and exposition are found; but when the datum is deficient, these also fail, since exposition and determination belong to the datum: for this will be the same with the proposition. Indeed, what else do we say, when determining in the aforesaid problem, unless that it is requisite to find an isosceles of this kind? But such was the proposition: if then the proposition has neither this datum, nor thing sought, exposition will, indeed, be silent, because there is no datum; but determination will be neglected, lest it should become the same with the proposition: but you may find many other problems of this kind, especially in arithmetic, and in the tenth book of these Elements, as, to find a medium comprehending two right lines commensurable in power, and every thing of this kind.
But every datum may be given in these four modes, either in position, or proportion, in magnitude or form; for a point, indeed, is given in position only, but a line and the rest in all the four. Thus, when we say, to bisect a given rectilineal angle, we declare the species of the angle given, as that it is right lined, lest we should also seek to bisect a curvilinear angle by the same methods. But when we say, from the greater of two unequal right lines, to cut off a part equal to the less, the lines are given in magnitude; for the less and the more, finite and infinite, are the proper predications of magnitude. But when we say, that if four magnitudes are proportional, they shall be also alternately proportional, the same proportion is given in the four magnitudes: but when it is requisite, from a given point to place a right line equal to a given right line, then the point is given in position. From whence, since position may be various, construction also receives variety; for the point is given either without the right line, or in the right line, and in the extremity, or without the extremity of the right line. Since, therefore, a datum has a four-fold acceptation, it is manifest, that exposition also is four-fold; but sometimes it connects two or three modes. Again, we find that demonstration sometimes possesses things proper to demonstration, exhibiting the thing sought for from mediate definitions; for this is the perfection of demonstration, but that sometimes it argues from certain signs. And it ought not to be concealed, that geometrical discourses have every where that which is necessary, on account of the subject matter, but are not every where perfected by demonstrative methods. For when, because the external angle of a triangle is equal to the two internal and opposite ones, it is shewn, that the three internal angles of the triangle are equal to two right, how is this demonstration from the cause? And is not a sign the medium in this case? For the external angle not yet existing, since the internal angles exist, they are equal to two right, since it is a triangle, though the side is not produced; but when, by a description of circles, the triangle, which is constituted, is shewn to be equilateral, the apprehension takes place from the cause. For we say, that the similitude and equality of the circles is the cause of the triangle’s equality with respect to its sides.
But geometrical discourses are likewise accustomed to make the conclusion, in a certain respect, two-fold. And this, when they exhibit things agreeable to the data, and reason universally, recurring from a particular conclusion to that which is universal; for when they do not use the property of the subjects, but placing the data before our eyes, describe an angle or right line, they think that which is concluded in this, is to be concluded in every thing similar: they pass on therefore to universal, lest we should think that the conclusion is particular. But their transition is effected in the best manner, since they employ, in demonstration, the things placed, not considered as such, but considered as similar to others: for it is not because such a particular angle is proposed that they effect a bipartite section, but because it is rectilineal only. But quantity, is indeed, proper to the proposed angle; but rectilineal is common to all right lines: let then the given angle be a right one. If therefore, we receive rectitude in the demonstration, we cannot pass to every species of right lines; but if we do not subjoin its rectitude, or being right angled, but alone consider its being rectilineal, the discourse may be adapted to all right lined angles; and all that we have previously observed we may contemplate in this first problem. For