:lxii DISSERTATION ON THE not, from the difficulty of understanding whether it is founded on peculiar or common principles; fince it is ne-ceffary that both these should be applied in the constitution of all real knowledge and science. * Again, axioms differ from poftulates in this :---they demand our affent without any previous folicitation, from the illuftrious certainty they poffefs. Their truth may, indeed, be denied by external fpeech, but never from internal connection. He who denies that equal things fhall remain from the fubtraction of equal, diffents, as Euripides fays, with his tongue, and not with his heart. But demonstration depends not on external speech, but on intellectual and internal conviction; and hence, axioms derive all their authority from intrinsic approbation, and not from public proclaim. For the prompt decifions of the tongue are frequently diffonant from the fentiments concealed in the fecret receffes of the heart. Thus the geometrician does not fpeculate thofe lines which are the objects of cor Axioms have a fubfiftence prior to that of magnitudes and mathematical numbers, but fubordinate to that of ideas; or, in other words, they have a middle fituation between effential and mathematical magnitude. For of the reasons fubfifting in foul, fome are more fimple and univerfal, and have a greater ambit than others, and on this account approach nearer to intellect, and are more manifeft and known than fuch as are more particular. But others are deftitu.e of all thele, and receive their completion from more ancient reafons. Hence it is neceflary (ince conceptions are then true, when they are confonant with things themselves) that there fhould be fome reafon, in which the axiom afleiting, if from equals you take away equals &c. is primarily inherent; and which is neither the reafon of magnitude, nor number, nor time, but contains all these, and every thing in which this axiom is naturally inherent. Vide Syrian, in A ith. Meta. p. 48. + cometry, indeed, wishes to fpeculate the impartible reafons of the foul, but fince the cannot ufe intellections deftitute of imagination, the extends her difcourfes to imaginative forms, and to figures endued with dimenfion, and by this means fpeculates immaterial reasons in thefe; and when imagination is not fufficient for this purpose, the proceeds even to external matter, in which the deferibes the fair variety of her propofitions. But, indeed, even then the principal defign of geometry is not to apprehend fenfible and external form, but that interior vital one, refident in the mirror of imagination, which the exterior inanimate form imitates, as far as its imperfect nature will admit. Nor yet is it her principal defign to be converfant with the in ginative form; but when, on account of the imbecility of her intelkation, he cannot receive a form deftitute of imagination, the fpeculates the immaterial reafon in the purer form of the phantafy; fo that her principal employment is about univerfal and immaterial forms. Syrian. in Arift. Meta. p. 49. poreal poreal fight, but fuch as are exhibited by mental conception, and of which the delineations on paper, or in the duft, are no more than imperfect copies, notes, and refemblances. Thus, when he draws a pedal line which is not pedal, or an equilateral triangle which is not equilateral, we must pay no regard to the defignations of the pen, but folely attend to the intellection of the mind; for the property demonstrated of fome particular line, is in the conclufion applied to one that is univerfal, and this true line could be no otherwife fignified to the learner than by a material defcription. The certainty of axioms is, indeed, in a measure obvious to every one. For what more evident than that nothing exists of which it is poffible, at the fame time, to affirm and deny any circumftance of being? Indeed, so illustrious and indubitable is the light of this axiom, that in any demonstration we are afhamed to affign it the place of an affumption. It would almost seem prolix and superfluous, fince there is nothing more manifeft and certain; and yet there are cafes in which it is neceffary to rank it among affumptions. And these take place whenever the intention is to conclude the existence of fomething as true, and of its opposite as falfe. Thus, for instance, in the demon-stration that the world is finite, we affume this principle, and then reafon as follows: Bound and infinite cannot be at the fame time affirmed and denied of any body: The world is a body: Therefore the world is not at the fame time finite and infinite. And in this genus of demonftration, the major propofition ought always to affimilate with the conclufion. But the above axiom is not the only one obvious, for the fol lowing lowing poffeffes equal certainty; that of every thing which exists, either affirmation or negation is true. This axiom is of great use in demonstrations leading to an absurdity; for he who demonstrates the impoffibility of any opposite affertion, neceffarily establishes his own. Hence it is we affirm that the diameter of a fquare is either commenfurable or incommenfurable with its fide; and this general principle is accommodated, and, as it were, defcends into its proper matter as often as that which it poffeffes of univerfal is contracted to a certain genus; for, as we have previously observed, common principles are not admitted in demonftration without any restriction; but then only when their general nature is limited to fome particular fubject, by which they become peculiar and appofite. 14. Wisdom, or the first philofophy and logic, agree in not using axioms after the fame manner as other arts; but on the contrary, they confirm and establish their certainty, though with this difference, that the logician reafons only from probabilities, but the metaphysician from the highest certainty and evidence. Befides, we do not Syrianus, in his excellent Commentary on Ariftotle's Metaphyfics, (which does not fo much explain Ariflotle, as defend the doctrine of ideas, according to Plato, from the apparent if not real oppofition of Ariftotle to their existence), informs us that it is the bufinefs of wifdom, properly fo called, to confider immaterial forms or effences, and their effential accidents. By the method of refolution receiving the principles of being; by a divifive and and definitive method, confidering the effences of all things; but by a demonstrative procefs, concluding concerning the effential properties which fubftances contain. Hence (fays he, becaufe intelligible cffences are of the most fimple nature, they are neither capable of definition nor demonstration, but are perceived by a fimple vifion and energy of intellect alone. But middle effences, which are demonftrable, exift according to their inherent properties: fince, in the moft fimple beings, norhing is inherent befides their being. On which account we cannot say that this is their effence, and that fomething elfe; and hence they are better than definition and demonstration. But in univerfal reafons, confidered by themselves, and adorning a fenfible nature, effential accidents fupervene; and hence demonstration is converfant with these. But in material fpecies, individuals, and fenfibles, fuch things as are properly accidents are perceived by the imagination, and are prefent and abfent without the corruption of their subjects. And thefe again being worse than demonftrable accidents, are apprehended by figns, not indeed by a wife man, considered as wife, but perhaps by phyficians, natural philofophers, and all of this kind rank rank logic in the order of the fciences, because it is destitute of some determinate genus or fubject, as it is neither converfant about lines, nor numbers, nor proportions. And its chief concern is about apparent properties, and not fuch as are effential to a subject. Hence, in logical difquifitions, we confidently employ interrogations, as equally fubfervient to the affirmation or negation of an opinion :---a method utterly impracticable, if we only employed those principles which are univerfally acknowledged; fince it is impoffible of the fame thing to prove contrary properties,---as of the foul, that it is mortal and immortal; but he who demonftrates, affumes one definite part of a queftion, because his purpofe is not to interrogate, but to trace out the latent paths of truth. And hence, if any one affirms that the foul is moved, and immediately after denies it, he is no longer a subject worthy the exercise of our difcurfive and reasoning powers. Again, it may fo happen, that the fame science at one time confiders why a thing is, at another only explains its existence, or that it exifts, without confidering the cause. Thus, the fyllogifm which concludes by mediate propofitions, demonftrates without affigning the proper caufe: but that which determines by immediate ones, in a great measure explains the caufe or reafon of exiftence. Thus, he who infers that trees do not breathe because they are not animals, reafons from a mediate and fecondary caufe, because there are many animals, fuch as infects, which exift without breathing: but he who infers this from their want of lungs, demonftrates from the immediate and primary caufe. Thus, the following fyllogifm is a mediate one, or such as requires one or more mediums to establish its certainty: Every thing that is not an animal does not breathe; A tree is not an animal; Therefore a tree does not breathe. i Here Here the major propofition is evidently mediate, because we are still to feek why that which is not an animal does not breathe, which the following immediate fyllogifm folves. Every thing that is not endued with lungs does not breathe ; Every thing that is not an animal is not endued with lungs; ergo, Every thing that is not an animal does not breathe. Again, the fame fcience may demonstrate the existence of a thing, or that it exifts, and the cause of such existence, as often as it affigns two immediate reasons; but the one from the proper caufe, the other only from a fign. Thus, he who demonftrates the increase of the moon, from the plenitude of her orb, infers the cause of such increase; but on the contrary, he who collects the plenitude of her orb from her increase, reasons only from a fign, and can alone declare its existence. And, indeed, it often happens that the cause and sign reciprocate, fo that as from the sign we advance to the caufe, demonftration from the caufe frequently recurs to the fign. Thus, from the breadth and firmness of the bafis, we collect the permanent duration of the pyramid; and from its extended existence we infer the frength of its fupport. Whenever, then, the argument originates from a sign, it gives evidence to the conclufion, as from fomething more known than its cause. When it begins from the cause, it proceeds from that which is first. in the order of nature, to that which is laft, and reasons as from the proper principle of the thing. Sometimes the cause and fign do not reciprocate. Thus, although wherever there is smoke, we infer the existence of fire; yet we cannot infer, that wherever there is fire fmoke exists. Thus, from the palace and the picture we collect the existence of the architect and painter; but the Jaft may exift without the firft;-the living architect with Out |