to what is useful in them, they ought only to be improv'd in those things, which have an immediate and sensible affinity with Arts, and the rest ought to be neglected as a Vain Theory. But this wou'd be a very wrong Notion. As for Instance, the Art of Navigation hath a necessary Connexion with Astronomy, and Astronomy can never be too much improv'd for the Benefit of Navigation. Astronomy cannot be without Opticks by reason of Perspective Glasses; and both, as all other Parts of Mathematicks, are grounded upon Geometry, and to go as far as you can, even upon Algebra.

Geometry, and especially Algebra, are the Keys of all the Inquiries, that can be made concerning Magnitude. These Sciences which are only conversant about abstruse Relations, and simple Ideas, may seem dry and barren, whilst they keep within the Verge of the Intellectual World; but mixt Mathematicks, which stoop to Matter, and consider the Motion of the Stars, the Augmentation of moving Forces, the different Passages of the Rays of Light through different Mediums; the different Effects of Sound by the Vibration of Things; to conclude all those Sciences, which discover the particular Relations of Sensible Magnitudes go on farther and more securely, when the Art of discovering Relations in General is more perfect. The Universal Instrument cannot be too extensive, too handy, or too easily apply'd: It is useful to all the Sciences, and they cannot be without it: And therefore among the Mathematicians of the Academy, who are design'd to be useful to the Publick, the Geometricians and Algebrists make a Class, as well as the Astronomers and Mechanicks.

However, it is certain, that Speculations purely of Geometry, or of Algebra, are not about useful things: But it is certain too, that those that are not, either lead or belong to those that are. It is in it self a very barren thing to know, that in a Parabola a Subtangant is double the corresponding Abscissæ; but yet it is a Degree of Knowledge necessary to the Art of throwing Bombs, so exactly as they can do now. There are not by far so many evident Uses as Propositions or Truths in the Mathematicks: Yet it is enough if the Concourse of several Truths is generally of some use.

Farther, a Geometrical Speculation, which was not at first applicable to any use, becomes so afterwards. When the greatest Geometricians in the Seventeenth Century set about to study a new Curve, which they call'd a Cycloide, they only engag'd themselves in a meer Speculation out of Vanity, striving to outdo one another by the Discovery of difficult Theorems. They did not even pretend that this was for the Publick Good; however by diving into the Nature of the Cycloide it was found, that it was destin'd to make Pendulums as perfect as may be, and carry the Measure of Time as far as it can go.

It is the same thing with Natural Philosophy as with Geometry. The Anatomy of Animals seems insignificant; and it only concerns us to know that of Human Body. But yet some Parts of it, which are of so nice, or so confus'd a Make, that they are invisible, are sensible and manifest in the Body of an Animal. Hence it is, that Monsters themselves are not to be neglected. The Mechanism conceal'd in a particular Kind or in a common Make, is unfolded in another kind, or in an extraordinary Make; and one wou'd be almost apt to say, that Nature by multiplying and varying so much her Works, can't sometimes forbear betraying her Secrets. All that the Antients knew of the Load-stone, was, that it attracts Iron. But whether they did not value a Curiosity, which promis'd them nothing; or that their Genius did not lead them to make Experiments, they have not examin'd this Stone as carefully as they might. One Experiment taught them, that it turns of its self towards the Poles of the World, and did put into their Hands the inestimable Treasure of the Mariners Compass. They might easily have made this Discovery important, and yet they did not do it; and if they had spent a little more time upon a Curiosity which seem'd useless to them, the Latent use of it had soon appear'd.

Let us always make a Collection of Mathematical and Physical Truths; happen what it will we can't hazard much by it. It is certain, that they shall be drawn from Springs, whence a great many useful ones have already been drawn. We have reason to presume, that we shall draw from thence, some that shall shine as soon as they are discover'd, and convince us of their Usefulness. Other Truths shall stay some time till a piercing Meditation, or some happy Accident discovers their Use. Some Truths being consider'd by themselves shall be barren, till they are consider'd with reference to one another. Lastly, let the worse come to the worse, some shall be eternally useless.

I mean useless with reference to sensible and gross Uses; for otherwise they shall not be so. An Object upon which alone you cast your Eyes is the clearer and brighter, when the neighbouring Objects, which however you do not look upon, are also enlighten'd; because it hath the Benefit of the Rays, which are reflected from them. Thus those Discoveries, which are palpably useful, and deserve our chiefest Attention, are in some measure enlighten'd by those, which may be call'd useless. For all Truths make one another more lucid.

It is always useful to have right Notions, even of useless Subjects. And tho' we cou'd reap no benefit by the Knowledge of Numbers and Sines, yet it wou'd still be the only certain Knowledge granted to our Natural Light, and they wou'd serve to give our Reason the first Habit of and Inclination to Truth. They wou'd teach us to operate upon Truths; to take the Thread of them, which is generally very fine and almost imperceptible; and to follow it as far as it reaches: In a word, they wou'd make Truth so familiar, that we might on other Occasions know it at first sight, and almost by Instinct.

A Geometrical Genius is not so confin'd to Geometry, but that it may be capable of learning other Sciences. A Tract of Morality, Politicks, or Criticism, and even a Piece of Oratory, supposing the Author qualify'd otherwise for those Performances, shall be the better for being compos'd by a Geometrician. That Order, Perspicuity, Precision and Exactness, which some time since are found in good Books, may originally proceed from that Geometrical Genius, which is now more common than ever, and in some manner is communicated by one Relation to another, nay even to those that do not understand Geometry. Sometimes a Great Man draws all his Cotemporaries after him; and he who hath the justest Claim to the Glory of having settled a new Art of Arguing, was an Excellent Geometrician.

Lastly, whatever raises us to Great and Noble Reflexions, tho' they be purely Speculative, afford a Spiritual and Philosophical Utility. The Wants of the Mind are perhaps as many as those of the Body. She desires to extend her Knowledge: All that can be known, is necessary to her, and there can be no better Proof than this, that she is design'd for Truth. Nothing perhaps can redound more to her Glory, than the Pleasure that is felt sometimes, in spight of ones self, in the dry and crabbed Questions of Algebra.

But without running counter to the common Notions, and recurring to Advantages which may seem too far fetch'd and refin'd, it may fairly be own'd, that the Mathematicks and Natural Philosophy have some things which are only subservient to Curiosity; and so have those Sciences which are most generally acknowledg'd to be useful, as History, &c.

History doth not in every Part of it supply us with Examples of Vertue and Rules for our Behaviour. For besides these, therein you have a View of the perpetual Revolutions of Human Affairs, of the Beginning and Fall of Empires, of Manners, Customs, and Opinions which continually succeed one another; and in a word, of all that rapid, tho' insensible, Motion that carries all before it, and incessantly alters the Face of the Earth.

Had we a mind to oppose Curiosity to Curiosity, we shou'd find that instead of the Motion, which