name. For its duty is to move the inherent knowledge of the foul; to awaken its intelligence; to purify its cogitation; to call forth its effential forms from their dormant retreats; to remove that oblivion and ignorance, which are congenial with our birth; and to diffolve the bonds arifing from our union with an irrational nature. It plainly leads us to a fimilitude of that divinity who prefides over this fcience, who manifefts intellectual gifts, and fills the univerfe with divine reafons; who elevates fouls to intellect, wakens them as from a profound fleep, converts them by enquiry to themselves; and by a certain obstetric art, and invention of pure intellect, brings them to a bleffed life. To whom indeed, dedicating the prefent work, we here conclude our contemplation of the mathematical science. corporeal nature. But by a contrary procefs, I mean, by applying mathematical fpeculations, . to experimental purposes, we shall blind the liberal eye of the foul, and leave nothing in its · Acad but the darkness of corporeal vision, and the phantoms of a degraded imagination. What Part Geometry is of Mathematics, and what the Matter is of which it confifts. IN N the preceding difcourfes we have confidered thofe common properties which respect the whole of the mathematical fcience; and this we have done agreeable to the doctrine of Plato; at the fame time collecting fuch particulars as pertain to our prefent defign. But confequent to this it is requifite that we should difcourfe on geometry itfelf, and on the propofed inflitution of the elements, for the fake of which we have undertaken the whole of the prefent work. That geometry then, is a part of the whole of mathematics, and that it obtains the fecond place after arithmetic, fince it is perfected and bounded by this, (for whatever in geometry may be expreffed and known, is determined by arithemetical reafons) has been afferted by the ancients, and requires no long difcuffion in the prefent enquiry. But we alfo may be able to relate our opinion on this particular, if we confider what place, and what effence its fubject matter is allotted The defign of the prefent chapter is to prove that the figures which are the fubjects of geometric fpeculation, do not fubfift in external and fenfible matter, but in the receptacle of imagination, or the matter of the phantafy. And this our philofopher proves with his ufual elegance, fubtilty, and depth. Indeed, it must be evident to every attentive obferver, that fenfible figures fall far fhort of that accuracy and perfection which are required in geometrical definitions: for there is no fenfible circle perfectly round, fince the point from which it is defcribed is not without parts; and, as Voffius well obferves, (de Mathem. p. 4.) there is not any sphere in the nature of things, that only touches in a point, for with fome part of its fuperficies it always touches the fubjected plane in a line, as Ariftotle fhews Protagoras to have objected against the geometricians. Nor muit we fay, with that great mathematician Dr. Barrow, in his Mathematical Lectures, page 76, "that all imaginable geometrical figures, are really inherent in every particle of matter, in the utmost perfection, though not apparent to sense; jult as the effigics of Cæfar lies hid in the unhewn marble, and is no new thing made by the ftatuary, but only is difcovered and brought to fight by his workmanship, i. c. by removing the parts of matter by which it is overshadowed and involved. Which made Michael Angelus, the most famous carver, fay, that foulpture was nothing but a purgation from things fuperfluous. For take all that is fuperfluous, (lays he) from the wood or stone, and the reft will be the figure you intend. So, if the hand of an angel (at least the power of God) should think fit to polish any particle allotted among the univerfality of things. For from a proper furvey of this, the power of the fcience which knows this fubject matter, the utility arifing from it, and the good acquired by its learners, will immediately appear. Indeed, fome one may doubt in what genus of things he ought to place geometrical matter, fo as not to deviate from the truth it contains. For if the figures concerning which geometry difcourfes, exift in fenfible natures, and cannot be feparated from the dark receptacle of matter; how can we affert that geometry frees us from fenfible objects, that it brings us to an incorporeal effence, that it accuftoms us to an infpection of intelligibles, and prepares us for intellectual energy? Where shall we ever furvey among fenfible objects a point without parts, or a line deftitute of breadth, or a superficies without profundity, or the equality of lines from the centre to the circumference; or the multangles, and all the figures of many bafes, concerning which geometry informs us? Laftly, after what manner can the reasons of such a science remain free from all poffible confutation; fince, indeed, fenfible forms and figures are fufceptive of the more and the lefs, are all moveable and mutable, and are full of material variety; among which equality subsists mixt and confused with its contrary incquality, and into which things without parts have proceeded into partition, and interval, darkened with the shades of matter, and loft in its infinite folds? But if the subjects of geometry are removed from matter, are pure forms, and are feparated from particle of matter, without vacuity, a spherical fuperficies would appear to the eyes, of a figure exactly round; not as created anew, but as unveiled and laid open from the difguifes and covers of its circumjacent matter." For this would be giving a perfection to fenfible matter, which it is naturally incapable of receiving: fince external body is effentially full of pores and irregula rities, which must eternally prevent its receiving the accuracy of geometrical body, though polished by the hand of an angel. Befides, what polishing would ever produce a point without parts, and a line without breadth? For though body may be reduced to the greatest exility, it will not by this means ever pafs into an incorporeal nature, and defert its triple dimension. Since external matter, therefore, is by no means the receptacle of geometrical figures, they muft neceffarily refide in the catoptric matter of the phantafy, where they fubfift with an accuracy fufficient for the energies of this fcience. It is true, indeed, that even in the purer matter of imagination, the point does not appear perfectly impartible, nor the line without latitude : but then the magnitude of the point, and the breadth of the line is indefinite, and they are, at the fame time, unattended with the qualities of body, and exhibit to the eye of thought, magnitude alone. Hence, the figures in the phantafy, are the proper recipients of that univerful, which is the object of geometrical fpeculation, and reprefent, as in a mirror, the participated fubfiftence of thofe vital and immaterial forms which effentially refide in the soul. fenfible "What Part Geometry is of Mathematics, and what the Matter is of which it confifts. IN N the preceding difcourfes we have confidered those common properties which respect the whole of the mathematical science; and this we have done agreeable to the doctrine of Plato; at the fame time collecting fuch particulars as pertain to our present defign. But confequent to this it is requifite that we should difcourfe on geometry itself, and on the propofed inflitution of the elements, for the fake of which we have undertaken the whole of the prefent work. That geometry then, is a part of the whole of mathematics, and that it obtains the second place after arithmetic, fince it is perfected and bounded by this, (for whatever in geometry may be expreffed and known, is determined by arithemetical reasons) has been afferted by the ancients, and requires no long difcuffion in the prefent enquiry. But we also may be able to relate our opinion on this particular, if we confider what place, and what effence its fubject matter is allotted * *The defign of the prefent chapter is to prove that the figures which are the fubjects of geometric fpeculation, do not fubfift in external and fenfible matter, but in the receptacle of imagination, or the matter of the phantafy. And this our philofopher proves with his ufual elegance, fubtilty, and depth. Indeed, it must be evident to every attentive obferver, that fenfible figures fall far fhort of that accuracy and perfection which are required in geometrical definitions for there is no fenfible circle perfectly round, fince the point from which it is defcribed is not without parts; and, as Voffius well obferves, (de Mathem. p. 4.) there is not any sphere in the nature of things, that only touches in a point, for with fome part of its fuperficies it always touches the fubjected plane in a line, as Aristotle fhews Protagoras to have objected against the geometricians. Nor muft we fay, with that great mathematician Dr. Barrow, in his Mathematical Lectures, page 76, "that all imaginable geometrical figures, are really inherent in every particle of matter, in the utmost perfection, though not apparent to fenfe; just as the effigics of Cæfar lies hid in the unhewn marble, and is no new thing made by the ftatuary, but only is difcovered and brought to fight by his workmanship, i. e. by removing the parts of matter by which it is overshadowed and involved. Which made Michael Angelus, the moft famous carver, fay, that foulpture was nothing but a purgation from things fuperfluous. For take all that is fuperfluous, (lays he) from the wood or fionc, and the reft will be the figure you intend. So, if the hand of an angel (at least the power of God) fhould think fit to polish any particle allotted among the univerfality of things. For from a proper furvey of this, the power of the science which knows this fubject matter, the utility arising from it, and the good acquired by its learners, will immediately appear. Indeed, fome one may doubt in what genus of things he ought to place geometrical matter, fo as not to deviate from the truth it contains. For if the figures concerning which geometry difcourfes, exift in fenfible natures, and cannot be feparated from the dark receptacle of matter; how can we affert that geometry frees us from fenfible objects, that it brings us to an incorporeal effence, that it accuftoms us to an infpection of intelligibles, and prepares us for intellectual energy? Where fhall we ever furvey among fenfible objects a point without parts, or a line deftitute of breadth, or a fuperficies without profundity, or the equality of lines from the centre to the circumference; or the multangles, and all the figures of many bafes, concerning which geometry informs us? Laftly, after what manner can the reasons of such a science remain free from all poffible confutation; fince, indeed, fenfible forms and figures are fufceptive of the more and the lefs, are all moveable and mutable, and are full of material variety; among which equality fubfifts mixt and confused with its contrary inequality, and into which things without parts have proceeded into partition, and interval, darkened with the fhades of matter, and loft in its infinite folds? But if the fubjects of geometry are removed from matter, are pure forms, and are feparated from particle of matter, without vacuity, a spherical fuperficies would appear to the eyes, of a figure exactly round; not as created anew, but as unveiled and laid open from the disguises and covers of its circumjacent matter." For this would be giving a perfection to fenfible matter, which it is naturally incapable of receiving: fince external body is effentially full of pores and irregula rities, which muft eternally prevent its receiving the accuracy of geometrical body, though polished by the hand of an angel. Besides, what polishing would ever produce a point without parts, and a line without breadth? For though body may be reduced to the greatest exility, it will not by this means ever pafs into an incorporeal nature, and defert its triple dimenfion. Since external matter, therefore, is by no means the receptacle of geometrical figures, they must neceffarily refide in the catoptric matter of the phantafy, where they fubfift with an accuracy fufficient for the energies of this fcience. It is true, indeed, that even in the purer matter of imagination, the point does not appear perfectly impartible, nor the line without latitude: but then the magnitude of the point, and the breadth of the line is indefinite, and they are, at the fame time, unattended with the qualities of body, and exhibit to the eye of thought, magnitude alone. Hence, the figures in the phantafy, are the proper recipients of that univerfal, which is the object of geometrical fpeculation, and reprefent, as in a mirror, the participated fubfiftence of thofe vital and immaterial forms which effentially refide in the foul. fenfible |