THE GREAT AND NEW ART OF WEIGHING VANITY: OR A Discovery of the Ignorance and Arrogance of the great and new Artist, in his Pseudo-Philosophical Writings. By M. Patrick mather's, Arch-Bedal to the University of S. Andrews. To which are annexed some Tentamina d● motu penduli & projectorum.  
GLASGOW, By ROBERT SANDERS, Printer to the City, and University, 1672.   

THE PREFACE TO THE READER.  
READER,  
I doubt not but thou art surprised to find me in print: and I assure you, that it is not more above your hope and expectation, than it is contrair to my former designs and resolutions: But as Atis his dumbness from the womb could not keep him from bursting into speech against those soldiers whom he saw ready to have killed his father; so my general insufficiency in all things else, cannot keep my natural affection in longer silence, when I see my bountiful Mother, this ancient and famous University, and all her beautiful Daughters, the other Universities of this Kingdom, in hazard to be murdered by one of their unnatural children.  
And finding that he with whom I have to do, hath given but a very lame and partial account of the occasion of our debate, I judge it both thy interest and mine, that I correct it by a more full, perfect and impartial one: For as ●he Magicians feigned miracles found greater belief with the Egyptians, than the true ones of Moses; so a false information having nothing to contradict it, oft times prevails as true with us.  
Thus than it is. My adversary having published his Tyrocinia Math. and his Ars. Magna & Nova, etc. one here who well understands those things, intending to oblige the Author, and redeem his Country from further injury by his writings, friendly represented to him some of his failings in them. And another, whose judgement he ought to have esteemed much, with the same intention, expressed to one of his nearest friends, his dislike of those Books, and his regrate for the loss which the Author put himself and his Country to by them. But this was not sufficient to convince him of his weakness; for he proceeds to give the world another instance of his folly, in printing his hydrostatics; and notwithstanding what had passed, he yet fancies that the Masters of this University have as high an esteem of his sufficiency, as he himself: And therefore not doubting of their encouragement to so noble a work, he confidently fends his petitory letters to some of them, entreating their own concurrence, and their assistunce for procuring the encouragement of others thereto.  
With his Letters, he sent this following Edict.  
Forasmuch as there is a Book of Natural and Experimental Philosophy in English, to be printed within these four months, or thereabout; Wherein are contained many excellent and new purposes: As first, Thirty Theorems, the most part whereof were never so much as heard of before: in which are proposed briefly the chiefest and most useful principles of that new Doctrine, anent the wonderful weight, force, and pressure of the water in its own Element. There are next, twenty Experiments in order to that Doctrine, not only most pleasant, and most easy to all capacities, but most useful likewise, which are set down after this method. First, each particular Experiment is briefly and clearly described, by its own distinct Schematism and Figure. Secondly, the curious Operations, and natural effects of it are showed. Thirdly, the true causes of these natural effects are searched into, and most evidently explicated, and demonstrated; not only by the force of reason, but by the evidence of sense also. And lastly, at the close of each Experiment, you will find most naturally deduced from the preceding Demonstrations, many excellent and new Conclusions (hitherto unknown) and these for the advancement of natural knowledge, and practice; among which, mention is made of a new and more commodious way of Diving. After all which, there is a number of Miscellany Observations; some whereof are Experiments made in Coal-sincks, for knowing the power of Damps, and ill Air, by killing of Animals. Some made for knowing the variation of the Compass here: and an excellent way for knowing, by the eye, the Sun or Moon's motion in a second of time, which is the 3600. part of an hour, and many others of different kinds, useful and pleasant.  
These are therefore to give notice to all ingenious Persons, who are lovers of Learning, that if they shall be pleased to advance to Gedeon Shaw Stationer, at the foot of the Lady's steps, three pound Scots, for defraying the present charges of the said Book, they shall have from him, betwixt the date hereof and April next to come, one of the Copies: And for their further security in the interim, the Author's obligation for performing the same. Edinburgh the 14. of December 1671.  
Which so exposed to my Masters the vanity of that confident man, that they were forced plainly to let him know their mind, as is expressed in the first Letter of his Postscript.  
To this he returned an answer, which, though it as little deserved his superlative commendation, as their censure, was abundantly discreet for obliging them to silence, until his Book should come to light. But to show how contrair to his nature this was, it quickly repent him of his discretion; and a little after, without any such provocation, as he alleges, he alarmed this place with a flood of his fury, whereof he dischargeth himself in the second letter of his Postscript.  
My Masters thought it unworthy of them to give any reply to this, lest by engaging themselves in a debate with one who had nothing wherewith to entertain them, except railing and calumnies, they had stained their reputation, and gained to themselves nothing but the name of foolish persons, for speaking to a fool in his folly: but I (to be ingenuous) having no much greater reputation for learning then himself, was content to hazard it against him: and knowing well his bragging humour to be such, as would make him insult and erect Trophies, if nothing were replied, I sent to him a Letter, which, to my best remembrance, was in the words following.  
Sir, I admire ex●eedingly the forwardness of your humour (I will call it no worse) in your last to—: he is a person not concerned in you or in your books, neither will he ignorantly commend any thing, as it seems ye expected he should have done, when ye sent him these papers. Ye might have known long ago, that he had no veneration for what ye had formerly published▪ for he made no secret of his mind, when he was put to it. Ye may mistake him, if ye think that any by-end will cause him speak what he thinks not: nevertheless he delivered your commission, and was willing to be inconcerned, expecting their answer. They pressed him to know his judgement of your last piece: he told ingenuously the truth, that there was none of them had less esteem for it then himself. He hopes ye are so much a Christian, that ye will not be offended with him for speaking what he thought, when he had a call to it; and yet, albeit ye seem to favour him more than others, he hath ground to look upon himself as one of the Sophistical rabble, for they only are such who condemn any thing ye do, the rest of the University continuing always learned persons. It is to no purpose to apologise for themselves, ye take all for granted, which ye have heard: I shall not put you to the pains of proving it; yet it seems ye would hardly have believed it so easily, had not your conscience told you, that they had some reason for their judgement, which really was this following: That they see nothing in your last piece, new and great, (albeit it be Ars nova & magna) save errors and nonsense; as your demonstrations of the Pendulum, your Nihil spatiale, your Gravitas circularis & horizontalis; your question, Whether or no a body may be condensed in a point? etc. too many to fill several letters: for ye must not call experiments new inventions, otherwise we are all making new inventions every day; neither must ye call different explications new inventions, else the same thing might be invented by almost every Writer. I admire how ye question the R. Society; for I desire to know one point of doctrine, which ye or they either pretend to, concerning the weight of the air, the spring of it, or any thing else in your book, save mistakes, which was not received by all Mathematicians, and the most learned of Philosophers, many years before any of you put pen to paper. Ye have been at much pains to prove that by experiment, which all the learned already grant, and some have demonstrat à priori from the principles of Geometry and Staticks, and many à posteriori from experience, if sense may be called a demonstration: Yet ye are the only man who produceth the Ars nova & magna, when all others are out of fashion. But more to your commendation, it seems ye do all these wonders by Magic; for ye have the ordinair principles of none of these Sciences: Euclid is as much a stranger, as reason in all your Books: and for this, Perue Mathematicos semper celebrabere fastus! At last ye come to prove a new doctrine, which before now was near 2000 years old, with thirty new Theorems, which must not be named, because they are of such a tender and delicate complexion, that the very naming of them will make them old. There are also many other excellent things, which will be all new when they were printed but yesterday. It is like, some of these days, we may have an Ars nova & magna, to prove that a piece of lead is heavier than so much cork. I know not wherefore ye undervalue any man, because he hath not as great esteem for your notions as yourself: Have not we as much freedom to speak our mind of you, as ye have to write yours of the R. Society, and the University of Glasgow? The greatest hurt ye can do us, is to make Dromo famulus one of our Principals. I think it not strange that ye using only demonstrations of sense, should admire the force of our imagination, in affirming no method of Diving so good as that of Melgim. I am sure that the man diving for a continual time, if he be not also of your invention, must breathe of the air; and this air must either be kept close by itself, as in Melgims' way, or communicate with the air above. If the latter be your invention, I doubt ye must also have some Chirurgical invention to apply to your Dyver at his return, if he go to any great deepness: If the former, it is the same with Melgims; and you cannot, neither any man else help it, but in circumstances (which altars not the method) and perchance to little purpose. As for Archimedes, I am sure he wanted no necessary requisite to prove the weight of water in its own Element. I know not what else ye intent to prove: always I am as sure that he had two great requisits, which ye want; to wit, Geometry, and a sound head. As to what ye write concerning the imperfections of Sciences; the scientisical part of Geography is so perfected, that there is nothing required for the projection, description and situation of a place, which cannot be done and demonstrat. The scientifical part of Optics is so perfected, that nothing can be required for the perfection of sight, which is not demonstrat, albeit men's hands cannot reach it; and these being the objects of the foresaid Sciences, your authority shall not persuade me, that it is altogether improper to call them perfect. In the hydrostatics, it were no hard matter to branch out all the Experiments that can be made, into several Classes, of which the event and reason might presently be deduced, as consectaries (I speak not here of long deductions, as ye seem to rant) to something already published: if it be noticed but rudely (as ye, not understanding what niceties of proportion means, must do) only considering motion and rest: And I believe there is none ignorant of this, who understands what is written in this Science. Upon this account writing to you, I might call it perfect; albeit I know there are many things relating to the proportion and acceleration of the motions of fluids, which are yet unknown, and may perchance still be. Ye shall not think that I speak of you without ground; for in your Ars magna & nova, ye bring in your great attempts for a perpetual motion; all which a novice of eight days standing in hydrostatics would laugh at. I do not question that this age hath many advantages beyond former ages; but I know not any of them, it is beholden to you for: only I admire your simplicity in this. Astronomers seek always to have the greatest intervals betwixt observations, and ye talk that ye will give an excellent way for observing the Sun or Moon's motion for a second of time; that is to say, as if it were a great matter that there is but a second of time betwixt your observations. I wonder ye tell me the eye should be added; for the invention had been much greater, had that been away, I do confess that a good History of nature is absolutely the most requisite thing for learning; but it is not like that you are sit for that purpose, who so surely believe the Miracles of the West, as to put them in print; and record the simple meridian altitudes of Comets, and that only to halfs of degrees, or little more, as worth noticing. However, if ye do this last part concerning Coal-sinks well, and all the rest be but an Ars magna & nova, ye may come to have the repute of being more fit to be a Collier then a Scholar. Ye might have let alone the precarious principles and imaginary wordless of Des Cartes, until your new inventions had made them so: For I must tell you, Des Cartes valued the History of Nature, as much as any experimental Philosopher ever did, and perfected it more with judicious experiments, than ye will by all appearance do in ten ages. Ye are exceedingly misinformed, if ye have heard that any here have prejudice or envy against you; for there is none here speaks of you but with pity and commiseration: neither heard I ever of any man who commended you for what he understood. As for your Latin Sentences, if they be not applied to yourself. I understand them not; for here we are printing no Books, we are not sending tickets through the Country to tell the wonders we can do: We are going about the employments we are called to, and strive to give a reason for what we say. Where then are our doli & fallaciae, tabulae & testes, sapientia ad quam putamus nos pervenisse? etc. In these things ye publish, ye know there is no Sophistry, but clear evidence: If ye had done such great matters in Universal & ens rationis, ye might have had a shift; but here ye must either particularise your inventions, or otherwise demonstrat yourself derogatory to the credit of the Nation: For what else is it to confound R. Societies and Universities with an Ars magna & nova; and yet when ye were put to it in print, to show your inventions, all ye could say was, that the publisher should have reflected upon the wisdom of the Creator, etc. so that the Poet said well of Democrites, etc. of which I understand not the sense, except ye make yourself the summus vir, and us all the Verveces. I suppose this may be the great credit that ye say ye have laboured to gain to your Nation; to wit, to get us all the hornable title of Wedders. No more at present, but hoping this free and ingenuous Letter shall have a good effect upon you (for I am half persuaded, that the flattery of scorners and ignorants, hath brought you to this height of imaginary learning) and that when ye come to yourself, ye will thank me for my pains. I rest,  
Your humble servant.  
After this I had no notice of him or his Book, until a copy of it came to my hands: which, when I had opened it, I found dedicat to a Noble Person; whose very name being there, did create in me a greater respect for the Book, than I thought myself capable of for any of the Author's works; and made me fear some finer things in this, than any other of his Books would suffer me to expect. For having known his Lordship an ornament to this Place, when his Virtue was but in blossom, I have easily given credit to that universal testimony, which reports him to have gained to himself an high esteem among Strangers, by those excellencies, which are the glory of his Family and Name; and therefore I could not but apprehend this present, offered to his Lordship on so solemn a day, to be something extraordinar.  
But having read over his Theorems, I admired the presumptuous arrogance of the Author, in concerning the authority of so Noble a Name in so worthless a trifflle: And having returned to the Dedication, to see what he said for himself, I justified his first application for Pardon, that he had prefixed his Lordship's Name to the bastle and abuse of a Noble subject. Then I considered the motives of the Dedication, and found them great; yea so great, that I wonder they did not fright him from so daring an attempt: For his Lordship, I hope, hath not given security to Strangers abroad, that he might draw upon himself injury from his Countrymen at home; his virtues have not made an Italian shelter under his Patrociny, that this bold Scribbler might be encouraged to send his Lordship through the world, as a Protector of falsehood, and countenancer of such as cannot handle truth without corrupting and defiling it. Could not his Lordship's Heroic virtues, and understanding mind; could not the learning and other excellent endowments of his Lordship's Father, Grandfather, and Great-Grand-father; could not the Dignity of their famous Ancestors, and the Antiquity of their Illustrious Family, preserve him from the importunity of this impudent man, who will needs enlighten his dark ignorance with the splendour of his Lordship's Name? Was not his Lordship's being an encouragement to learning, sufficient to have kept this arrogant pretender there o, from soliciting his Lordship's authority, to his folly and infirmity? Surely, when he addressed this Book, he either little considered his Lordship's abilities to judge thereof, or else he intended to court his friendship and affection, for a defence against the power of his understanding; & if he gain his design, he hath reason to say, that his Lordship's goodness is proportioned to his other accomplishments.  
After this view of the Dedication, I went through the rest of the Book unto the Postscript, where I find mention made of the Letter which I sent to the Author, who was wiser than to print it, lest thereby he had published his own shame; but he lets it not pass without a cast of his craft: For finding that by it his ignorance is discovered, he foams and rages, he is troubled in spirit, because he is disturbed in the exercise of his Art; that is, because he is not permitted to call other men's truths, his own, and his own falsehoods and follies, rare and useful truths, and obtrude them upon the world as such; and being fettered with that reason which opposeth him, he, in the bitterness of his spirit, vomits out his spite against her, calling her Sophistry, Nonsense, and whatever his anger suggests to him: and breathing nothing but revenge, he calls together his choicest virtues Fury, Malice, and Boldness; and having got them to join with his Ignorance, he endeavours by these united forces, to uphold his cause: Nor was any of them wanting to him, as may appear from their particular achievements, which are remarkable in that review of my Letter, which summeth up his Postscript; and in sum, equally betrays his Insufficiency and Insincerity. For therein he treateth the Masters of this University so unworthily, (as he had done in the second Letter of his Postscript, in answer to that Gentleman, who, by direction, wrote unto him their mind) that I know nothing like it, except the spirit of its Author, and that entertainment which he in the Preface to his Ars magna, and pag. 472. gives to the late Archbishop of Glasgow (who had been most kind to him) and Masters of the College there, in which some than were, & yet are, who may be his teachers in any thing he pretends to.  
But this Postscript doth not sufficiently discover the Author's vortues, and therefore he spends a part of his first Epistle to the Reader, in such flat and vulgar rail, as prove him fitter for nothing, then to hold the principality among the Street-scolders. And moreover, that the provocation may be complete, he gives a formal appeal to any who dare state himself his adversary: and makes such ostentation of his strength and courage, that, rather than want a combatant, he will purchase one with gold; for he offers a Guiny for every Theorem which shall be everted, either in this, or his last Book. And such is his generosity, that I cannot doubt, but he will also be as noble in requiting the labour of any, who shall give him some Tyrocinia, whereby he may correct his discovered errors.  
Sure I am, there may be as much gained here as would tempt my Adversary once again, to blot a great many sheets of paper, if to boot, he could be assured of a Crown, or Rix-dolar, or (rather than lose his market) a Legged-dolar, for every Book that should stand himself no more than two Marks.  
Now, Reader, I am confident thou thinks me further engaged after all these provocations, then that I can retreat with honour; and so think I myself: And therefore I have accepted my Adversaries Challenge. I have examined all his Books: I have weighed them in the balance of reason, and have found them so light, that they deserve no better name than Vanity. I have displayed the Author's infirmity and folly in every one of them, without other design then to protect my Country, and particularly all such as he endeavours to concern in his Writings, from the mean thoughts and misapprehensions of those who have no other character of both, than they receive from them.  
Yet in this Review I have not displayed all the enormities of this Arrogant pretender to Knowledge; for this should have made my Book swell as far above a just measure, as his Arrogance and Insolence is above every thing, except his Ignorance; seeing, every period of his Writings is either pregnant with falsehood; or if it contain a truth, which he hath taken from some other, his probation thereof is either from false principles, or management so silly and childish, as makes it appear ridiculous. Neither have I taken notice of all the impertinencies whereof he is guilty, lest thereby I had hazarded the reputation of my good nature: But I have only exposed some of his grosser failings, to let the world know, that he hath not so much wit, as himself presumes; and discovered his inveterat malice, to undeceive those who think him a man of much sincerity.  
And this I have done with so much evidence and demonstration, that I fear not thy censure, if thou be intelligent: Not have I sent this book to your hands, under any other Patricony, then that of Reason; for she is able to recommend it to the favour of my Friends, and protect it from the Fury and Malice of my enemies. But if it were not, that the meanness of my person and station should have made my address as indecent, as the naughtiness of my Adversaries Present made his, I would have offered it (as a testimony of my humble duty, and sincere respect) to that Noble Person, to whom he hath dedicat his hydrostatics; and as earnestly have solicited hisVnderstanding to judge of my Truths, as my Adversary hath done his Lordship's Friendship to accept, his Favour to protect, and his Name and Authority to convoy his Falsehoods through the world. Nor should I either have precipitated or suspended my address for finding so craving an opportunity, as the day of his Lordship's Birth and Majority.  
From my Chamber in S. Andrews, the 24. day of July 1672.   

THE GREAT AND NEW ART OF WEIGHING VANITY.  
AS in combating, each party first intends his own defence, and in the second place only prepares an assault for his Antagonist: So I, before I make any attempt on my Adversaries other Writings, shall endeavour to wipe off that dirt which he hath thrown upon me, in the Postscript and Preface to his Hydrostatics.  
I think it no wonder that my Adversary hath suppressed that Letter of mine, which he mentioneth in his Postscript, and I have printed in my Preface; for this gives him the greater liberty to belie it; which he doth most splendidly, when he saith, that it is full of barbarous rail, passing all bounds of civility against himself, friends, and works: (whereas there is not a word of his friends in it: and what is therein said of his Works, the following Treatise will manifest, if it deserve the name of barbarous rail.) Nor is it strange to see one who wants truth on his side, make lies his refuge: But he may henceforth look for the common infelicity of liars, not to be believed, if he shall chance to stumble upon truth.  
I had reason to fall upon his Ars magna, etc. because I judge ex ungue Leonem, or rather, ex cauda Catum. Nor should the bare title have been passed by, because it is arrogant and false, as shall be made to appear in its own place. I am unjustly in this compared to blind Vejento; for he had the beast but at one hand; but to whatever hand I turn me, I find the beast there. And because my Adversary complains, that I have only snarled at the horse heels, I shall henceforth endeavour to pull the Ass from the saddle.  
I excuse my Adversary for not interpreting his Latin verses, because they were sent him from— without interpretation.  
I am obliged to his esteem, in supposing me a Master in an University. He was never judged worthy of that dignity here: and by his ingratitude to Glasgow, he hath proven himself unworthy ever to have had it there, or any where else. And I wonder, that judging me a Master here, he should think strange that I am not so Pedantic, as (in imitation of him) to stuff my Letter with Latin Sentences altogether impertinent to our debate; and which in his Letter, and his review of mine, serve for nothing so much as to express his malice and virulency. Yea, there be two things which I think more than strange inconsideratness in him. The first is, that he accuseth me for not writing pertinent language in my Mother tongue; whereas in the very next page he writes, He hath done as the Ape did, that thrust the Cat's foot into the fire, because he durst not do it himself; whereof, if he or— make good sense and Grammar, I shall give him back one of those Guinies which I am to have for everting his Theorems.  
The other is, that he should challenge an Vniversity-man for writing a Letter without a Latin Sentence, whereas he hath written Volumes of Mathematics, without ever (for any thing I have yet seen) citing a Classic Mathematician, except once Euclid Prop. 24. lib. 1. El. Geom. in the 265. page of his hydrostatics, and that erroneously. For Euclid hath two sides in one triangle equal to two in another, and our Author hath only one side in each triangle. This is like the Tarsel of a Mathematician.  
I had reason to ask, Where are our doli & fallaciae, tabulae & testes, sapientia ad quam putamus nos pervenisse? For, first, none here being further concerned then in answering his importunate Letter, desiring the Universities encouragement for printing his hydrostatics; how could any so much as dream, that a man in his right wits, should provoke others to overthrow the title of a Book Tabulis & testibus, after he had once refused to let them know any part of what was contained in the Book? And yet this Author hath done it, as he himself testifies in the 310. page. Sure no other would, for this dexterous wit is peculiar to him. But good Sir Sciole, let me tell you, it had been as great wisdom, either still to have concealed your great knowledge, or else to have kept up your provocations, whereby you should have saved me from the trouble of producing proof & witness against you, and yourself from the shame of being convicted guilty of both Ignorance and Insolence by them: For I assure you, that before your Indiscreet Challenges, I had no design to expose the folly of your arrogant pretences, and the contemptible infirmity of your acquittances, otherwise I might have drawn very lucky instances of both from your Ars magna & nova, etc.  
Secondly, before he charged upon the Masters here his doli & fallaciae, there was nothing which could be a ground for it, seeing all that had passed, was his Letter desiring their concurrence to the printing of his Book, and their answer, wherein they declare their mind with much candour and calmness. And he tacitly acknowledgeth the injustice of his challenge, in answering my question from that Letter in which the question itself is contained: For it is against both Reason and Religion, first to calumniat, and then to justify the calumny from something posterior thereto; and it is yet the worse in him, that he doth it by an untruth, in alleging my letter to have another design, than any, except himself, can discern; nor would he see it, if any other thing could be found to excuse his malicious reflections upon persons of known integrity.  
Thirdly, there are none among those whom he reproacheth, who have been so long at his School, as to learn either arrogantly to pretend to the knowledge of those things to which they are strangers; or vainly to fancy themselves knowing in that whereof they are ignorant.  
After this, my Author proceeds in such a strain as would almost proyoke Meekness herself to make a satire. But it is so pitiful, that it cannot excuse a serious answer from being impertinent; and therefore I pass it, without suffering myself to digress into Satiric reflections upon his vanity therein. Only I beg his liberty, that since he hath made me the Cat, I may henceforth, without offending him, catch the Rat as oft as he comes in my way.  
Now my Adversary susficiently animat with rage, prepares himself for making a furious assault upon some passages of my letter, about perfection of Sciences, and begins it very learnedly, by bringing in the Historical part of Geography, as a part of the Science of Geography; which is as good Logic, as if he had said, that black is a part of white, because they are both colours. But that he may the better understand this, I tell him, that Geography simpliciter is not a Science: for a great part of it is only History: and I cannot suppose him so ignorant, as not to know that Science and History (albeit all learning, as almost all things else, receive their denomination from the most noble part) are very different: Especcially in Mathematics, where the scientifical part is firm and Geometrical, and the Historical part subject to the weakness of our senses; the one consisting in Methods and Demonstrations, the other in Practices and Observations. All these things he here mentioneth, and thousands more, can be done by sure and Scientifick Methods, and therefore are perfected in so far as they are a Science; except only the measuring the height of the Sea above the Earth; and this I think can only be done by himself, to whom it is easy to make Rivers run upwards, and so to work many wonders in hydrostatics. I am sure that any person who understands Logic, will find by these, that my Adversary hath triumphed before the victory, and hath unjustly called my argument a Fallacy, while he had only reason to call it a Caption, since he was catcht thereby.  
He next falls upon the Optics, where after he hath vapored a little, to no other purpose then to display his Pedantry, and discover his dislike of modest expressions, he asketh a question which proves him a stranger to this part of learning. But that he may reap some instruction from this debate, let him know that the Optics hath scientifically so far perfected the sight, that it demonstrateth this Theorem: In all Telescops, as the focus from the eye glass is to the distance of the focus from the object glass, so is the simple appearance of the object to the appearance of the same through the Telescope. And therefore if the distance from the focus to the eye glass be one inch, and the distance of the focus from the object glass 100000, the object will appear 100000. times longer or broader by help of the Telescope, then to the simple eye: Or with this Telescope you may see as well at 100000. miles' distance, as with the simple eye at one: If the glasses (or rather mirrors, because they lose no rays, and have caeteris partibus, all one determinat reflection) be sufficiently large, and of the true Geometrical figure. By the same method, the demonstrative, or scientifick part, teacheth us to see at any finite distance, as if it were three foot or less. The like consideratis considerandis, is true in Microscops and Scotoscops also. If our Author do question this rule, he shall find it in Escinardi Optica, and in the Philosophical Transactions, page 4005. as also in others before them both. It is like if he had known it, he had spoken better sense in his New Optical experiment.  
He is mistaken in saying, that it is not known how the sight is made; for it is done by bringing all the rays coming from one point of the visible always to one point of the retina. It was never motioned by any learned man (since the Optics came to this perfection) that any brutes yet known, should see otherwise then men: Fishes indeed, because of the dense medium they live in, have their crystalline rounder; and night-beasts, such as Cats and Owls, their uvea larger: yea, many other particulars there are, of which the Optics do evidently demonstrat the reason.  
Our Author might have remembered since he was a Professor of Philosophy, that lights and colours are qualities, at least according to him; and therefore not the object of any Mathematical Science, which is always quantity.  
Reflection and Refraction were fully handled by Des Cartes; for it is out of doubt, That the angle of incidence is equal to the angle of reslexion, and the sins of the angles of incidence proportional to the sins of the refracted angles. Infraction, is the same with Refraction, and therefore impertinently repeated.  
It is no wonder the Lord Verulam was not of my mind; for he died before the time of Des Cartes, who brought the Optics to this perfection. But it is no small wonder to find a man pretending so highly to learning, as our Author doth, and yet print himself a stranger to the progress thereof.  
It is true indeed that M. Newtown hath discovered an inconvenience in Refractions, which was not formerly known, and that therefore Metallin Mirrors are more proper than glasses: but this hath not added any thing to that universal rule I presently mentioned, which scientifically bringeth the sight to any degree of perfection, and holdeth in these Telescops, as well as in all others: yea, these Telescops were known before, only their advantage above others was not known.  
What he saith of M. Hook, is most improper: seeing there he only promiseth to accomplish or bring to practise what hitherto hath been attempted, or by all most desired; not at all mentioning the Science, which our Author questions.  
Let any man consider the vast extent of that rule, and think what can be more large. I do not question that there may be many excellent and subtle inventions for promoting sight, as to practise: but I am sure the scientifick part cannot make the sight infinitely perfect, and it hath already brought it to any degree of finite perfection.  
He flatters himself that he hath gained the victory, as to the hydrostatics: but upon what account, may be seen in my Letter; which being written in private, only for dissuading him from making himself ridiculous, and for curing him of his blind presumption, was framed to his capacity, and not for the learned world. And seeing it was necessary, because of the importunity of his Letters, to signify to him, that this Science was already perfected, as to all these things whereof he is capable; it was more civility and respectfully spoken, to say, that the hydrostatics were already perfected, then to say, that they were further perfected than he could reach.  
Our Author should know that all mixed Mathematical Sciences, are nothing else but Geometrical Demonstrations, founded upon some Physical Experiment: So that Geometry, to speak properly, is the only Science in Mathematics, and their only store-house for rules, methods, reasons and inventions: It is certainly defective in several things; but these are far above our Author's conception.  
He next strives to persuade the unlearned, that he hath first taught Astronomers the use of Telescops and Pendulum clocks; but I leave this to the examination of his experiments. Yet I must not pass that which he desires the Reader to mark; to wit, my nonsense, in saying, That the invention of representing the Sun or Moon's motion in a second of time, had been greater, if the eye had been away. And I entreat the Reader to mark as well, how M. Sinclars dulness maketh him impute his own nonsense to me: for in his printed Letter Feb. 22. he challenged as a great neglect, that the Eye is not added in an expression of a former Letter; as if any could have dreamt that the observation might be without the eye; to which I answered, That the invention had been greater, if the eye had been away: and surely so it had: Nor could this have escaped M. Sinclar, if he had not wanted his eyes; but his blindness hath made him stumble upon my expression: and because he could not bruise it with his fall, he hath lashed me for his own fault. Surely this discipline is very near in kind to his doctrine, for they are both unreasonable.  
I have nothing to say against his miracles in the West, especially that grand one of the Sun seen in Winter for an hour about midnight, eight degrees above the Horizon: except, that it is only mentioned in his Book; no man, I ever spoke to, having heard of it; although I know many who have been in the place mentioned, and very inquisitive concerning it. Besides, that laying one aside, it far surpasseth all miracles of the heavenly bodies, recorded in facred History.  
If our Author think that he was well exercised, when he was making his observations of the Comet, he should judge a part of his time well spent, in letting the world know for what they served: but he seems to intend no more; then to make men believe, that he is not ignorant of a degree or a minute, although he reckons the Sun's motion by inches.  
I question not, that a Coal-hewer is more useful to the Country than he and I both: and therefore he is obliged to me, for giving him a more useful trade, than he now driveth. Nor can I deny, but he justly deserved it; for a Coal-hewer is one who maketh gain by digging in another mans mine; and so hath he done; for that History of Coal which he hath printed, is none of his, although he hath made advantage by the publishing and sale thereof. But this is no great wonder, since the most part of the truths contained in his writings, are digged out of other men's works. And that the Author of this History may not escape the fate of others with whom he maketh so bold, he mixeth with his doctrine, some mistakes of his own, and particularly that erroneous application of Euclid above mentioned in page 4. of this Book.  
Now my Lords and Gentlemen, who are Coal-masters, I pray you consider how unjustly M. Sinclar infers, that I design for you no better name than I have given to him; and how maliciously he thereby endeavours to create in you a prejudice against me. I highly esteem and honour all such whose knowledge and virtue maketh useful, and ornaments to their Country. But pardon me, that I suffer not M. Sinclar to usurp to himself the name of a Philosopher for writing this History, (although it were his own) since he wants the Science of Coal; for it is not History, but Science, that makes the Philosopher.  
I need not concern myself much in his censure of Des Cartes; for he is as far exalted above my commendation, as he is without the reach of M. Sinclars detracting tongue.  
He may well say, that he is not afraid I shall come the length of his labours in Glasgow College, about Vniversale, and Ens-rationis; for in his last Logic Notes, he hath thirty sheets of paper upon Genus and Objectum Logicae, Vniversale and the predictables; which falsifies the first sentence of the Epistle to the Reader of his Ars Magna.  
He might have holden his peace of Rhetorical and Algebraical composition and resolution; for he knows no more of either but the name. If he had read this part of my Letter right, he would have had some other fancies, than he here expresseth; as I should show, were not this too sheepish a subject to be insisted upon.  
It is true that a Letter was sent to M. Sinclar, containing the words which he printeth; but it is as true, that the same Letter contained the condition of that promise which he there mentioneth; to wit, If he made it appear that his Book were answerable to his Edict. The concealing of this is so great a proof of his candour and ingenuity, that infallibly it will procure credit to any thing he affirms.  
Now this Good Man having spent many of his spirits in this tempestuous conflict, is oppressed with drowsiness; and having fallen asleep, he dreams all the rest of his Postscript. For I am sure there is not one in this University, who ever either had his name in an Almanac, or craved any man pardon upon such an account.  
I have seen the Pamphlet he speaks of with the Advertisement to the Reader, and found nothing in it of any ingenious Gentleman Artist, set upon inhumanely as by two Mastiffs; but some Printer checked for playing the Astronomer unhandsomely, and that under a borrowed name, for to make his Prognostication the more vendible; a practice too ordinar. Our Author here talking of two, judgeth this business to be of the same difficulty with that of D. Moor's butter Scon, which could not be sufficiently fenced from the violence of the Air, by less than the Syllogistical force of two bold brethren.  
However, if there be any errors in that Almanac, he bewrays his ignorance in passing them; while he lets a fling at the mistake of a Table, and at some Chronological Rhymes, things of no importance. For the first, it may be imputed to a piece of rashness, occasioned perhaps by the obscurity of that Tables explication, but not to ignorance; seeing such trifles, as Tobacco-boxtables, and Pocket instruments, which produce nothing, but what can be better done without them, conduce not to knowledge: And therefore no reproach for a man to be ignorant of them, being contrived only for Mechanics, and such sensible Demonstrators as my Adversary is. As for the Rhymes, I suppose there is as little necessity of thinking the Author of them, and of the Almanac, to be the same, as of judging the new and unheard-of Hydrostatical Theorems, and the bundle of Latin Sentences in the reply to my Letter, to have been tursed by the same hand▪  
I have no regard for Rhymes, and yet for recreation, I must take notice of our Authors two Criticisms; whereof one is, the two last lines exceed the former in a foot, contrare to that of Horace, Primum ne medio, etc.  
Consult our English Poets, Sir, what weight this authority hath with them. The other is: It should not have been said, Since that of nought the Lord created man. But, Since that of dust, etc. Pray you, Sir, is this sound Philosophy; and if it be, how taught you your Scholars, Cap. 7. de Causalitatib. Caus. Prim. Creatio est actio causae primae, quâres primo ex nihilo producuntur? But who then can this Prognosticor be? It is very probable, from the rabble of Astrology, (for there is none of that profession among us) that he is my Antagonists Apocalyptical Astrologue, who Lib. 6. Dial. Phys. 3. Sect. 1. besides his Astrological Predictions, and Prophecies out of the Old Testament, did from the Revelation of S. John, with great zeal declare many, and these even wonderful things, concerning the number of the Beast 666. and the Alphabetical letters A. B. I. S. of great affinity with it. The mystery of these must not be revealed, lest it occasion the discovery of that divine Astrologue.  
There is little heat here about Ens rationis; that crack-brained knave hath evanished, together with his Cousin-germain M. Sinclars dearly beloved Forma substantialis materialis. For aught I know, they have got in to his Nihil spatiale, to erect a College of Fanatic Philosophers.  
I Am now to examine his Epistle to the Reader, where he complaineth exceedingly of Envy, because the Masters of this University would not take his word for the novelty of his inventions: Nevertheless he must grant (if he will be ingenuous) that they have done him a courtesy, in causing him prefix a more modest Title to his Book, than his Edict carries.  
He wrongs M. Boyl egregiously, in causing him say generally, that Archimedes' Demonstrations have more of Geometrical subtlety than usefulness; whereas he saith only (in the Preface to his Hydrostatical Paradoxes) that many of his Hydrostatical Propositions have more of Geometrical subtlety than usefulness. It were nonsense to speak so of Demonstrations, seeing their only use is to prove the thing in question: which if they do, they cannot be called useless; and if they do it not, they cannot be called Demonstrations.  
Our Author now compares his method with that of Archimedes's forsooth. He is more speculative, our Author is more practical. So may a Trone-lord say: Archimedes was more speculative in his Staticks, and he more practical. Next Archimedes' Demonstrations are Geometrical, and his Physical. That is to say, Archimedes' reasons are sure and solid, and his are conjectures: And then Archimedes' Demonstrations are but for the use of a few, and these for the use of all. He might truly have added, And for all uses, except to convince; which is the proper use of a Demonstration. As for his last comparison, Archimedes was more wise then to illustrat that in his Book, which any mean man might do, and was already demonstrated. But our Author needs not imagine, that a rational man will venture any surprising Demonstration to the world, without practising it, if he can: yet there was no necessity that he should swell his Book with it. I say the like of Stevinus, in whose Demonstrations, I am not afraid our Philosopher show any defect, nevertheless that he be pleased to speak at random.  
He beginneth now to tell the strange things he hath invented. And first, he saith, that he considereth the pressure of the water with the pressure of the air jointly. Can our Author be so ignorant, that he knows not the arise of the Toricellian experiment? Was it not from the consideration of Pumps and other Hydrostatical machine's, that they had no effect above 33. or 34. foot? Was it not considered here by Galilaeus, that water pressed water no further than its own level; and it was probable, the weight of the Air might press it up the rest of the way (seeing it was not much) which it ascended in the Pump? Upon this account, he projected the experiment first in water, (where was considered the pressure of Water and Air jointly) and afterwards Toricellius perfected it in Quicksilver, judging rationally, that the great weight of the fluid by shortening the tube, would facilitat the experiment. In M. Boils continuation of Physico-Mechanical Experiments, Exper. 13. 14. 15. Doth he not consider the pressure of both together? Yea, is there any intelligent man who now speaks of a Pump, or any Hydrostatical engine, without considering both these pressures together?  
All these counterposing, which he speaks of, have been tried by M. Boyl, and also many more; to wit, oil of Turpentine, and oil of Tartar, etc. but if our Author please, he may try it yet with Ale, Beer, Urine, etc. and all these shall be new Experiments. He should have been more general in these trials, and more particular in the mysteries and secrets of the Art which he hath discovered, and none else can get notice of. Archimedes asserts the weights of all fluids in general, and consequently of the Air, if it be a fluid, which the Learned never yet denied: Yea, Archimedes' Commentator Rivaltus (who died long before the Toricellian experiment) mentioneth the Air and its weight.  
That assertion of M. Boyl is true at present, and will constantly be so, suppose every man alive print such Volumes as our Author hath done. However, the learned Doctor Wallace hath published a Book not long ago, notwithstanding all our Author's invention; in which he deduceth more than ever our Author shall know of the hydrostatics, as consectaries from one proposition.  
Now, Reader, I stay no longer here to consider my Adversaries indiscreet rail and provocations; for this were unworthy both of you and me: But that you may know, that I am a man of my word, I proceed to the survey of his works, as I promised in my Preface. And I am not a little encouraged to this, by the hope of gaining as many Guinies, as may help that pitiful poverty, wherewith he upbraideth me.  
But lest he think that the Problem which his Brother proposeth concerning the bringing up from the bottom of the Sea, any weight that can be sunk therein, hath bougled me, I think fit to give thee here three several answers thereto.  
First then, for effectuating that which is there proposed, you shall take the new invention, called, The Diving Ark, one so large that it requires a greater weight to sink it down, than the Pondera demersa: which being sunk down near to the Pondera demersa, the Dyver must first bind them to the Diving Ark, and then loose away the weight which did sink it: Now the Pondera demersa, being ex hypothesi, lighter than the weight which was sufficient to keep the Ark at the bottom, must of necessity be pressed up with the Ark by the water: and the nearer it cometh to the brim, the motion will be the swifter, not only for the acceleration of the motion, but also because the Air dilateth itself, and (as I determinat in my Examination of this dyvink Ark) the Ark is pressed upward with as much force, as the quantity of water equaling the included Air, would cause by its weight in the Air. But if the Inventor will take my word upon it, his Ark must be stronger than a Wine glass, and without holes in the bottom: nay, it must not have a Glass window of a foot in square, at least not near the bottom. And if the Pondera demersa be great, when he hath done his utmost, in case the bottom of the new Invention get out, you may have supply from the old hydrostatics: Thus,  
You shall take at a low water, some great strong tuns banded with iron, so many of them, that being all full of water, they are heavier than the Pondera demersa in the water; that is to say, that the weight of all these tuns full of water, may weigh more than the Pondera demersa, having rebated from their weight, the weight of their quantity of water. These tuns being all emptied and exactly closed, and iron chains or strong ropes tied to their iron bands, let the Dyver go down in his Bell, and bind these chains or ropes (all the tuns may be fastened to one chain) to the Pondera demersa, as near as may be; and the rising water shall lift the Pondera demersa from the ground; which being once done, they are easily drawn any where. If the Pondera strike on the ground, at the next low water stint the chains as much as ye can.  
I suppose any man who trieth these ways, will be best pleased with this, which hath been known these many ages: seeing it is far easier to multiply tuns, then to make a vast bulk of an Ark, with a bottom proportionably strong, to resist the pressure of the water, and to be troubled with a weight sufficient to demerge the same. These two Answers I have got from my two brethren the inferior Bedals, who are as fertile in affording satisfactory answers, as my Adversaries Brother is in starting subtle questions. If it be objected against the last of these two Methods, that it can only be practised where the sea ebbeth and floweth, I give you a third.  
Take two ships (any of which is sufficient to raise the Pondera demersa) the one deep loadned with stones, or any such thing, the other altogether empty. Bind the loaded ship as near as may be to the Pondera demersa (which may be easily done by the help of the Diving Bell) and then liver her into the other which was empty: This livered ship shall raise the Pondera demersa from the ground, which afterwards may be easily drawn any where. And if perchance they strike on the ground in the drawing, let them be bound again to the new loaded vessel, doing as formerly. This method, I suppose, you will find in Vitruvius, who is a very old Writer; and yet if M. Sinclar had given it, it is like, he would have listed it amongst his new Inventions, as he did Riccioli's erroneous argument against the motion of the Earth.  
Hitherto I have been employed in parreing those thrusts which M. Sinclar gives in at me, through all the Postscript, & part of the Preface to his hydrostatics: It is now high time for me to prepare an assault for him, this being a part of my Province: and in forming it, I shall make use of no weapon, but Reason: hoping from it, better success, than my Adversary hath had; & the rather, because he is so great a stranger to it.  
The first shall be upon his hydrostatics, because that began the debate. The second upon his Ars nova & magna, because of the reproaches my Masters have sustained for their just censure of it. And the last assault shall be upon his Tyrocinia, which indeed is more blameless than the rest, being freest from errors, and more consonant to its title; yet albeit it had no name prefixed, it could not but sufficiently discover the Tyro and the Great and New Artist, to be all one. All this shall be done in the proper language of each Book, that every work, & its examination, may be understood by the same Reader: And so I begin with the hydrostatics.   

AN EXAMINATION OF M. SINCLAR'S hydrostatics.  
Non equidem hoc fludeo, bullatis ut mea nugi●  
Pagina turgescat, dare pondus idonea fumo.  
Secreti loquimur:—  Pers. 
THat I had sufficient reason to quarrel the offer of thirty new and unheard-of Hydrostatical Theorems, shall appear from the examination of this Treatise; whereof all that is true, (for a considerable part of it is false and ridiculous) is the same with the doctrine of Archimedes and Stevinus, in the following Propositions: only our Author's doctrine is more loose, and less precise.  
As for what he hath written concerning the Bensil of fluids, generally applied, is false; seeing no Bensil hath hitherto been perceived in any fluid, except Air. And seeing the doctrine of the spring of the Air, is called by most of Authors, and particularly by M. Sinclar himself, Aërostaticks: I think not myself obliged to reduce it to the writings of Archimedes and Stevinus, who wrote only hydrostatics properly so called: yet in that subject also, (where he speaks truth) I shall in its due place trace him in Aërostatical Writers extant before him.  
In the review of this Tractat, I shall, for my hires sake, begin with the Theorems; and afterward take notice of a few things in the Observations and Experiments.  

§. 1. The Theorems reviewed, whereof a great part are proven false, others ridiculous, and the rest not new.  
I Shall here at once discover the falsity and ridiculousness of a considerable part of our Author's Theorems, and reduce the rest to these following Propositions of Archimedes and Stevinus.  


Archimedis Positio 1.  
Ponatur humidi eam esse naturam, ut, partibus ipsius aequaliter jacentibus & continuatis inter sese, minus pressa à magis pressa expellatur. Vnaquaeque autem pars ejus premitur humido supra ipsam existente ad perpendiculum, si humidum sit descendens in aliquo aut ab alio aliquo pressum.   

Prop. 2.  
Omnis humidi consistentis atque manentis superficies Sphaerica est, cujus centrum est idem quod centrum terrae.   

Prop. 5.  
Solidarum magnitudinum quaecunque levior humido fuerit demissa in humidum manens, usque eò demergetur, ut tanta moles humidi, quanta est partis demersae, eandem quam tota magnitudo gravitatem habeat.   

Prop. 6.  
Solidae magnitudines humido leviores in humidum impulsae, sursum feruntur tanta vi, quantò humidum molem habens magnitudini aequalem, gravius est ipsâ magnitudine.   

Prop. 7.  
Solidae magnitudines humido graviores demissae in humidum, ferentur deorsum, donec descendant: Et erunt in humido tantò leviores, quanta est gravitas humidi molem habentis solidae magnitudini aequalem.   

Stevini Postul. 3.  
Pondus à quo vas minus altè deprimitur, levius; quò altiùs, gravius; quò aeque altè, aequipondium esse.   

Prop. 5.  
Corpus solidum materiae levioris quam aqu●… cui innatat, pondere aequale est tantae aquae moli, quanta suae parti demergitur.   

Prop. 8.  
Corpus solidum in aqua levius est quam in aëre, pondere aquae magnitudine sibi aequalis.   

Prop. 10.  
Aquae fundo horizonti parallelo tantum insidet pondus, quantum est aquae columnae cujus basis fundo, altitudo perpendiculari ab aquae superficie summa ad imam demissae aequalis sit.    
Now, Reader, consider well these Propositions: my Author's Theorems; and my Censure, which is this.  
His first two are no Theorems; but only Suppositions. And the third, a sort of a definition, or rather, aliquid gratis dictum.  
The fourth, as he wordeth it, is false: for a broad fluid counterpoyseth more than a narrower; seeing a cylinder of Mercury one inch thick and twenty-nine inches high, counterpoyseth a cylinder of Air of the same thickness, and of the altitude of the Atmosphere: and one two inches thick with the former height, counterpoyseth four times as much Air. As he explicateth it, it is true, and the same with Archimedes' second Proposition; for the Demonstration holds, suppose ye divide the fluid by several pipes, if they have intercourse.  
Here he maketh a mystery of a very easy thing: for one pillar of water being ten times thicker than another of the same height, and consequently an hundred times heavier, hath no more effect than the other; for because of its base, it hath an hundred times as much resistance. And it is most clear, that if the resistance be proportional to the pressure, the effect must constantly be the same.  
His fifth, is a part of Archimedes' first position.  
His sixth also; for Archimedes' expulsion hindered with equal resistance on all sides, he calleth, Pressure on every side. I suppose he will hardly affirm, that this lateral pressure was not known before him; seeing Stevinus doth demonstrat, how much it is upon any plain howsoever inclining, in his Prop. 11. 12. 13. which our Author cannot do yet; at least, there is nothing in his Book either so subtle or useful.  
His seventh is the same with the last part of Stevinus' third Postulatum.  
The eight is manifestly false, (if fluids have a Bensil, as he supposeth, Prop. 17. 19) which I demonstrat from his own figure thus. The first foot E having one degree of weight, and the second foot I having equal quantity or dimension, and being lower than E, must have more weight; (according to his 17.) let it therefore have 1½ degrees of weight: then the weight of both these must be 2½. Now the third foot N, being of equal quantity with I, and lower, must (according to his 17.) have more gravity than it hath; (to wit, 1½) let it therefore have 2. degrees; and then the weight of all three is 4½ degrees: but 1. 2½, 4½, are not in Arithmetical progression; and therefore the Theorem is false.  
I must take notice, that if our Author had understood so much as the terms of Art; he would have said, The pressures of fluids are in direct proportion with their profundities. His inference there concerning a Geometrical progression is false; for there are many Geometrical progressions more than 1, 2, 4, 8, etc. And it may be in many several progressions, albeit it neither be in Arithmetical nor Geometrical progression. And, suppose he had not contradicted himself, his Theorem is evident from the 10. of Stevinus: For, according to it, the weights or pressures of fluids are equal to the weights of respective Cylinders upon the same, or equal bases; but the weights of such Cylinders are in proportion with their quantities, which is the same with the proportion of their altitudes.  
The ninth and tenth (as he explicateth himself) are only this, That fluids press upon bodies within themselves, and press up bodies lighter than themselves in specie; which is the same with his 6. and 13. The first of which we have examined already: and the other we leave to its own place. But what ground he hath for his sensible and insensible gravity, I shall discuss in the examination of his Ars magna & nova, which is all built upon this wild notion.  
His eleventh is manifestly false, as I shall afterward demonstrat from his own principles: for the Cylinder acquireth only a greater base, (our Author must understand that an Horizontal surface is the base, and sustains the pressure) and consequently a greater resistance, which maketh the same weight of less effect. It is evident that a weight of lead cannot press two foot in square, so much as one: yea the pressures of the same weight are always caeteris paribus in reciprocal proportion with the surfaces they press; as it is known by all Mathematicians, except only such pitiful ones, as our Author.  
The twelfth is evidently false; for, if ye take a bladder, or any tender vessel half full of water, and put the sides of it together, the fluid shall be moved from the unequal pressure of the vertical surface.  
The one half of the thirteenth is a part, but a very small one, of Archimedes' seventh, and vl: The other half is also a small parcel of Archimedes' sixth.  
His fourteenth is so much as he understands of Archimedes' fifth, and Stevinus' fifth.  
The fifteenth, seventeenth and nineteenth are false; unless the fluid have a spring, or be heterogeneous; none of which he hath made out: but if it were made out, the thing is obious, and noticed by M. Boyl in the thirty-sixth Experiment; yet only in the Air, which is known to have a spring.  
His sixteenth is ridiculous; seeing we see daily fishes, little particles of earth, horse hairs, and many other such bodies betwixt the surface and bottom of the water. Yea by adding a sufficient quantity of lead to a body lighter in specie then water, it may be made practicable: and is demonstrat both by Archimedes and Stevinus, supposing the water homogeneous; the contrair of which, our Author hath not yet made out. And more, even a body considerably heavier in specie then water, beaten out thin and broad, especially if it be concave below, may be suspended for a considerable time betwixt the surface and bottom of the water, providing it be laid parallel to the Horizon. But passing by all this, his method is unpracticable, and supposeth, without proving any thing, that water can suffer any degree of compression; and stones, lead, with other bodies, none at all.  
His eighteenth is the same with Archimede's seventh, and Stevinus' eighth.  
His twentieth is the same with his seventh, otherways he grants it not exactly true.  
His twentyone (as he wordeth it) is most manifest from that Statical demonstration I mentioned: For seeing pressures of the same weight are in reciprocal proportion with their resistances, and the resistances or resisting surfaces can be diminished in infinitum; it is evident that the least weight can produce any pressure, whether the heavy body be fluid or solid. But he explicateth himself otherways, relating to the spring of fluids, which is not yet proven in any fluid, save Air; and besides this, the Theorem is ridiculous, seeing the spring of any part (where all are equally pressed) is equal to the spring of the whole: for one pound weight presseth one foot as much, as two pound presseth two; and even so in any spring.  
His 22. and 23. are made manifest by Pecquet in his fourth Experiment, and M. Boyl in his 19 Physico-Mechanical Experiment, yea throughout all that book and many others, constantly calling the weight and spring of the Air divers, and yet bringing them both in for that same effect.  
The 24. is ridiculous; seeing it is true and obvious in all things, if there be no penetration of bodies.  
The 25. is evidently false, seeing waters upon the tops of hills support less, and in valleys more. Yea Doctor Wallace showeth in his Mechanics, pag. 728. that the Mercury both in M. Boils Baroscop, and his, fell sometimes at Oxford below 28. inches, and other times above thirty, and in the page 740. he mentioneth unquestionable experiments of 34. 52. and 55. inches. The contrair of this Theorem is also evident from many of our Authors own experiments, if any man think them worthy the looking over. And suppose he had hit right, this is nothing but the old Toricellian Experiment.  
His 26. is imperfect; first, seeing he speaketh only of fluids to be pressed up, it being also true in all other bodies. Secondly, he doth not determine how far the sphere of activity reaches; and yet all this is easily done and demonstrat from Stevinus his 10. For the body is pressed up, till it together with the fluid betwixt it and the bottom (not regarding what else intervene, but reckoning all for fluid) be equal in weight with a column of fluid, whose height is the same with the height of the fluid, and its base the same with the base of the former fluids portion, or equal to it: and besides all these, this is not different from M. Boils eleventh Paradox.  
His 27. is to say, that a pound of wool weigheth as much, as a pound of lead.  
His 28. is the same with that which he would say in the 4. and is true also in solids; if ye speak only of columns: For two unequal columns of the same height and matter press equally, seeing their resistances are proportional with their weights. In fluids (as I said already) it is the same with Archimedes' Second.  
His 29. might have been more general, to wit, That there can be no motion in fluids, without an unequal pressure: And then it had been the same with Archimedes' first position.  
His 30. is also a part of Archimedes' first position. For seeing pressure is judged only by expulsion the effect of it; and the expulsion is always caused where the least resistance is, which may be in a crooked line: wherefore then is not pressure also in crooked lines?  
His 31. is the 10 of Stevinus. Here again he justleth with that great difficulty, which I discussed in the 4. and telleth there is no way to answer, but his.  
In his 32. the Pondus & Potentia, are to say in plain Scots, a pressure and a resistance. He hath told in his 5. that in all fluids there was a pressure; but now it comes in his head, that a man may fancy a pressure without a resistance; & therefore he must guard against that. I suppose here, that his definition of the Staticks is new; otherwise the Tron-lords are the greatest professors of it.  
His 33. is to say, that there must be a motion, when the pressure is greater than the resistance; which is yet a part of Archimedes' first position, and never doubted of by the greatest ignorants.   

§. II. The Authors last Theorem, for its good service, examined by itself.  
NOw let us examine his last Theorem, which certainly should be the utmost reach of his wit; and therefore I will examine it more narrowly.  
First, let his two fluids in aquilibrio be, Water the one, and Quicksilver the other, The natural weight of Water being 1. the natural weight of Quicksilver is 14. Therefore according to his Theorem; as 1. the weight of the one is to 14. the weight of the other, so is the height of the one, to wit, Water, to the height of the other, to wit, Quicksilver: and therefore the Quicksilver should be 14. times higher than Water, which I leave to be determined by experience. He should have said, as the natural weight of the second, is to the natural weight of the first: Or rather, that their altitudes are in reciprocal proportion with their weights, or in direct proportion with their levitieses.  
Secondly, then in his progress, he saith, That by what proportion the one liquor is naturally heavier or lighter than the other, by that same proportion the one Cylinder is higher or lower than the other: here insinuating, that the weights and levitieses of two bodies are in the same proportion; and yet their proportions are reciprocal, and that is to say, just contrair: or otherwise, he must take the heights proportional with the weights, and the lowness with the levitieses; which are both false. At last, when he comes to his example, he makes the heights proportional with the levitieses, which I grant to be his meaning; but this showeth an intolerably confused wit.  
Thirdly, even this being granted, I shall demonstrat, that it doth contradict almost all his Theorems. And to that purpose, I assume these two Postulata.  
Post. 1. Fluids which have their weights or pressures proportional to their profundities, can have no Bensil: For if they have a Bensil, their pressure is not proportional to their profundities, (as I did demonstrat at his 8. Theor.) which is against the hypothesis.  
Post. 2. Quicksilver or water, have their weights and pressures in proportion with their altitudes. At least, so far as any man yet hath made trial; as M. Boyl witnesseth in the first Appendix to his Paradoxes: yea, our Author affirmeth it of all fluids, in his 8. Theor. and many places of his Experiments. The demonstration follows.  
Here upon the surface of the Earth, let the height of a Cylinder of Mercury be A, its weight, or the weight of the Cylinder of Air counterpoising it B, the height of this Cylinder of Air C. Also let the same Cylinder of Mercury be lifted up some distance from the Earth, and the Mercury will fall, so that the Cylinder of Mercury is now lower, whose height we call D, and weight, or the weight of its counterpoising aërial Cylinder E, the weight of this aërial Cylinder F; let the proportion betwixt the weights of Mercury and Air be as G unto H. By our second postulatum, A is unto D, as B is unto E; and by this 34. Theorem, H is unto G, as A is unto C: and also H is unto G, as D is unto F; and therefore A is unto C, as D is unto F; & permutando, A is unto D, as C is unto F; but A is unto D, as B is unto E; And therefore B is unto E, as C is unto F; and consequently (by the first Postulatum) the Air hath no Bensil; which is contrair to many of his Theorems, and all his Experiments.  
This destroys all his methods of measuring the height of the Air, Clouds, and Atmosphere, both here and in his Ars magna & nova. He might have known this mistake many years ago; for M. Boyl rejecteth this proportion betwixt the altitudes of the Air and of the Quicksilver in his 36. Physico-Mechanical Experiment, upon the same account. This letteth our Author see, that if fluids have no Bensil, his Theorem was obvious, and known to all.   

§. III. The Authors great skill in Dioptrics, examined.  
IN his third Observation, he maketh himself exceedingly ridiculous. For, first, he showeth hot how much the Telescop required, should magnify.  
Secondly, he showeth not how far the Telescop should be drawn out for this effect; for that draught which serves for a distinct and clear sight, will not serve exactly to project an Image; seeing sight requireth always parallel, or diverging rays, and the projection of an Image, converging.  
Thirdly, he seemeth to attribute the magnifying of Telescops to their length and goodness of the glasses; and yet there may be the best glasses imaginable placed in their due distance in a tube of 50. foot long, and not do so much as an ordinar tube of 5. inches; and yet both the glasses may do wonders with others which give them their due charge.  
Fourthly, he requires both the glasses to be very good, and there is no excellency required but in the object glass.  
Fifthly, he speaks of the Image, as if it were both near to the Tube, and far from it; and yet it hath one determinat place, the draught of the Tube never being altered, which he never once mentioned.  
Sixthly, he speaks of the Image of the Sun, that it is the more distinct, the nearer the glass; and yet this brightness near the glass, is nothing but a confused concourse of rays.  
Seventhly, when he hath observed his inches, he reduceth them not to degrees, minutes, or seconds, etc. for the Sun's motion is not reckoned in inches.  
Lastly, suppose he had done all these things aright; this method hath been ordinarily practised above these thirty years: Let him look Hevelij Selenographia, Scheineri Rosa Vrsina, and Doctor Wallace in the end of his Arithmetica infinitorum.  
It is here to be observed, that these Authors by such observations designed not to render the Sun's motion sensible to the eye. (which our Author values so much, and by some here was formerly called ridiculous) but only to observe its spots together with their motion, or else its eclipse: noticing only by the way, that swift motion of the Sun's Image, which was troublesome, and constrained them oft to alter the position of their Telescop.   

§. IV. Our Author's new Diving Ark, put to trial.  
THere is nothing in which our Author is more mistaken, then in his Diving Ark; for in all his discourse, he not only contradicts himself, (which is ordinar, and no great matter) but also the general doctrine of the hydrostatics. I shall therefore, to undeceive his Reader, demonstrat, That his Diving Ark sustains precisely as much pressure under water, as if it were hung in the Air with as much water in it, as now it hath of Air, rebating only a small matter which the compressed Air in the Ark weigheth. I do it thus.  
In his own figure, pag. 179. let PQ be the sufface of the water within the Ark, PY the distance of that surface from the upper horizontal surface, NIGH the distance of the top of the Ark from the upper surface. According to his 7. Theorem, the pressure is equal at P and at 4; and therefore according to his 8. Theor. seeing the water hath no sensible spring, the pressure at N without the Ark is to the pressure at P, as YN to YP; therefore the pressure at PQ, overcometh the pressure without the Ark at EH, by the pressure of a column of water, whose base is PQ, and the altitude HQ; but the pressure at EH within the Ark, wanteth only the weight of the column of Air PQHE, to make up the pressure at PQ; therefore the pressure within at EH, exceedeth the pressure without at EH, by the weight or pressure of a column of water, whose base is PQ, and altitude QH, abating the weight of the column of Air PQHE; Which wa● the conclusion to be demonstrated.  
I demonstrat this conclusion, supposing no man within the Ark; but if a man be there, it holds only of the Air about him, taking the man to be equal in weight with so much water. I would gladly know if our Author now would affirm, that, suppose the Ark were no stronger in the sides than a wine glass, yet it might go down 40. fathom without hazard, and that it may have a glass window a foot in square, and holes in the top, wherein ye may put your little finger: Yet I shall help him in one particular; There is more hazard in the first three fathoms, for the bursting or leeking of the Ark, then in the next three hundred, seeing the space filled with Air groweth less. Are these the great matters, which our practical Mathematicians invent, whilst others are nibbling at petty demonstrations?   

§. V. The honourable M. Boyl vindicated from our Author's ignorant censure, in his Exper. 17.  
I Resolved only (having considered the extraordinary pains it would take to examine all the nonsense, contradictions, absurdities, and superfluities in his Experiments and Observations, which almost every page is filled with) to take notice of these he mentioned in his Edict: but seeing him so bold, as (in his 17. Experiment) to insult over that learned Gentleman M. Boyl, I must, by permission of more learned Pens, which this great man's vindication doth deserve, undertake to demonstrat the truth of what M. Boyl affirmeth: that is to say, That the water REF (see the Authors fig. 24.) weighed in the Air, is of the same weight exactly, which it hath weighed in the water, according to M. Boils method. I do it thus.  
By my former Demonstration, before the water EFR enter the glass, the glass PR, is as much pressed upward in the water, as it would be pressed downward in the Air by its fill of water, rebating the weight of the Air now within it: Therefore the weight which keepeth the glass PR, in aequilibric in the water, must be the same with the weight of its fill of water in the Air, substracting the said weight of Air. Now when the water OF entereth, the glass PR is as much pressed upward in the water, as it would be pressed downward in the Air by EPF, full of water, rebating the weight of the Air EPF, which is the same with the former: and seeing at first the pressure of the glass upward, was equal to the weight of all PR, full of water rebating such a weight, and now the pressure is only equivalent to the weight of the water EPF, rebating the same weight; the pressure of it is now diminished by the weight of the water ERF: but the pressure is likewise diminished by the weight put in the scale O; and therefore that weight is equal to the weight of the water ERF, in the Air; Which was the conclusion in question.  
All that our Author speaks against this, is to no purpose. First, he saith, that though lead casteth the balance; but that cannot be, seeing the lead was there, before the balance was casten. He concludeth, That water doth press in water, but not weigh in water: I will not call this nonsense, but only retort, that upon the same account, Air will not weigh in Air; and yet I believe, he thinks, that he hath weighed Air in its self. It is like, he may say, that this is done by the Toricellian tube, where the air is exhausted: so might M. Boyl have said, that is in a glass bubble, where the water is exhausted: And I may also say of this whole Hydrostatical doctrine, that it is exhausted also, and can be no longer, without prejudice, kept back from its grave.  
THis waterish doctrine hath passed off with more credit than it deserved, having gasped out its last by virtue of that noble name, The Honourable Robert Boyl. I doubt not, Reader, but by this time thou art made weary by it; and so am I. Wherefore unwilling to return, and rake up its ashes, to thy further annoyance and mine, I shall go forward to the Ars nova & magna, and quickly show thee what novelty and greatness is there, without any prefacing; having no other testimony for it, than what is due to the rest of its fellow-works.    

EXAMEN ARTIS NOVAE ET MAGNAE GEORGII SINCLARI.  

CAP. I. Tres primi Libri Dialogorum Philosophicorum, & duo de Instrumentis Hydragogicis examinantur.  

§. I. Hic rejicitur Authoris Theorema primum.  
LIb. 1. 2. & 3. de Baroscopij phaenomenis agitur: quod Baroscopij vocabulum, sicut & quaedam alia, se primum excogitasse gloriatur Author, Regiamue Societatem plagij accusat, acsi ea è suis Manuscriptis compilaverit; licèt res ipsae jampridem extiterint.  
Dial. 1. lib. 1. Varia proponuntur theoremata, quorum primum (quod tantummodo divisio est, quam membrorum definitiones sequuntur) sie se habet. Quod corporafluida, uti Aqua, Aër & Hydrargyrus, duplicem videantur habere gravitatem, unam Sensibilem, aliam Insensibilem. Eam appello (inquit) gravitatem Sensibilem, quam sentio, dum verbi causa, amphoram aquae plenam, à terra manu allevo & sustollo; quod quidem omnium aliorum corporum gravium, etiam est proprium. Gravitas Insensibilis, fluidorum solummodo proprium, est illa vis & potentia quâ corporaseipsis leviora sursum pellunt, etc. Virtute hujus, ait Author sub finem Sect. 2. Circumsusum hunc aërem aequipondium efficere cum Hydrargyro, vel aqua, adminiculo tubi in forman Cylindri redacta. Sed haec definitio nullatenus convenit isti potentiae, quâ aër cum aqua tubi constituit aequipondium, nam virtute ejus, aër aquam (quae est corpus gravius) in tubo sursum pellit. Nec convenit aquae; nam haec in tubo, corpus levius viz. aërem deorsum premit aut pellit. Sidicatur, quod aqua tubi aërem prius quidem deorsum premit, sed sic premendo, eundem etiam necessario sursum pellit in locum cadentis aquae: Respondebitur, Obeandem rationem, lapidi cadenti gravitatem istam insensibilem similiter competere; quam tamen, supra assertum est, fluidorum esse propriam.  
Theorema tertium Sect. 5. est falsum; nempe, Aqua & id genus alia corpora fluida in libra naturali pendentia, gradatim insensibilem deperdunt gravitatem, prout gradatim reclinatur tubus vel Siphonis crus horizontem versus. Hoc fundamentum est totius doctrinae, lib. 1. & 2. Dialogorum Philosophicorum, & duobus libris de instrumentis Hydragogicis, traditae; praeterea, id passim fere praesupponitur in plerisque corporum fluidorum phaenomenis, per reliqua authoris opera solvendis. Qualis sit illa doctrina, hujus Theorematis eversione apparebit.   

§. II. Theorema praedictum fundamentale de sluidorum gravitate Insensibili evertitur.  
JUxta nostri authoris doctrinam, praesertim lib. 1. de Instr. Hydr. dial. 2. & lib. 2. dial. Philos. Dial. 3. Sect. 2. Hydrargyti cylindrus 29. digitis altus aequiponderat cylindro aëris eandem cum ipso basin habenti & altitudinem eandem cum Atmosphaera. Hinc, in ordine ad sequentis propositionis demonstrationem, hoc colligo postulatum.  
Postul. 1. Gravitatem insensibilem esse eam, quâ Hydrargyri v. g. Cylindrus dictae aëris columnae aequiponderat: & proinde, quò major est haec aëris columna, eò major est etiam Hydrargyri gravitas insensibilis; ita ut, si una aeris columna, alterius sit dupla, tripla, etc. erunt item cylindrorum hydrargyri illis aequiponderantium gravitates insensibiles una alterius dupla, tripla, etc.  
Alterum postulatum deducitur è lib. 2. dial. 1. Sect. 10. & lib. 2. dial. 2. dial. Philos. Sect. 5. ubi praeter alia, haec habet verba. Nam inde inferre licet, si divinâ providentiâ aëris altitudo augeretur; cylindri mercurialis altitudinem in Baroscopio similiter, servatâ nimirum proportione, majorem evadere. Et si eâdem providentiâ, ejus altitudo minueretur, minorem etiam Hydrargyri altitudinem fore. Hinc inquam colligitur postulatum 2. quod sic se habet.  
Postul. 2. Cylindrorum hydrargyri aequales bases habentium, gravitates insensibiles sunt in directa altitudinem proportione. Nam cylindrorum hydrargyri aequales bases habentium, duplò altior, aëris aut aquae duplo aequiponderat, & triplus triplo, etc. Quod etiam patet ex Sect. 8. & 9 Dial. 1. lib. 1. Dialog. Phil.  
Nunc sequitur propositio demonstranda Theoremati praedicto contradictoria.  
Sisint duo tubi 29. digitis hydrargyri repleti, aequè longi, & aequè crassi, superiori orificio occlusi, quorum alter sit ad horizontem rectus, viz. tubus DF, & alter ad horizontem 
Fig. 1.  reclinatus; viz. tubus DB; Dico utrumque tubum aequalem habere gravitatem insensibilem.  
Producatur cylindrus DB usque ad KH, ut sit ejusdem altitudinis cum DF, ducatur etiam QB ad angulos rectos cum DK, & CH, quae aequalis erit basi DG, propter aequalem cylindrorum crassitudinem. His factis.  
Est insensibilis gravitas cylindri DF, ad insensibilem gravitatem cylindri DH; ut DG, ad DC, ceu QB, ad AB. Etest insensibilis gravitas cylindri DH, ad insensibilem gravitatem cylindri DB; ut DK ad DE, seu, ut AB ad QB. Ergo ex aequalitate ordinatae, Erit insensibilis gravitas cylindri DF, ad insensibilem gravitatem cylindri DB; ut QB ad QB: h. e. sunt aequales. Quod erat Dem.   

§. III. Causa erroris in praecedenti Theoremate ab Authore commissi detegitur.  
QUoniam hoc theoremate fretus, in utroque volumine, ubicunque de fluidis fir sermo, passim fere hallucinatur Author; non ab re duxi, erroris in hoc theoremate fabricando originem detegere, quae est haec. Observavit, quod idem tubus eadem mole hydrargyri aut aquae repletus, versus horizontem reclinatus, aequipondium cum aëre non constituat, sicuti fecerat, dum fuit erectus: inde putavit ille, ejusdem hydrargyri pressuram in subjectum aëra debiliorem esse in situ obliquo, quam in recto: nequaquam animadvertens eandem vim aut pressuram manere posse aequalem, licet ob incrementum resistentiae de novo adveniens, minus quam antea efficacem: ut res se habet in hoc casu. Nam hydrargyrus tubi dum est erectus, premit in basin cylindri aërei circularem, suae basi circulari aequalem: cum verò tubus est reclinatus, idem hydrargyrus premit in basin cylindri aërei Ellipticam basi tubi circulari majorem, ideoque in majorem aëris cylindrum, quam tubô existente erectô. Ac proinde non mirum, si eadem pressura cylindri hydrargyri, tubô existente reclinatô, non possit adaequare resistentiam majoris cylindri aërei ad constituendum aequipondium, sicuti resistentiam minoris cylindri aërei sibi aequiponderantis adaequavit, dum fuit tubus erectus.  
Novi objici solere, fluidum magis inniti interioribus tubi reclinati partibus, quam erecti; ac proinde non aequè premere in subjectum aëra, in utroque situ. Sed Respondetur, hoc argumentum nihil facere contra praecedentem Demonstrationem; nam quantò magis premuntur pàrtes tubi reclinati interiores circa CB, quam totidem partes erecti; tantò minùs inde premuntur partes interiores circa DA, quam totidem tubi erecti partes: (quae omnes semper in eadem altitudine aequaliter premuntur.) Et consequenter, totus tubi mercurius simul sumptus, aequaliter innititur tubo, in utroque situ: seu quod idem est, quantò debilitatur gravitas, seu vis deorsum pellens cylindri mercurialis, ob reclinationem lateris CB, tantò etiam debilitatur resistentia, seu vis sursum pellens mercurij stagnantis, ob inclinationem lateris DA: ut optimè illustrat D. Wallisius in sua Mechanica, pag. 717.  
Per hydrargyri gravitatem insensibilem, nihil aliud quam pressuram, quod ad aequipondium cum aëre externo, nunc magis nunc minùs efficacem, ob resistentiam nunc minorem nunc majorem intellexit hic auctor: sed quia nunc majoris nunc minoris in hac pressura efficaciae rationem ignoravit, quando scilicet tanta sit, quanta ad constituendum aequipondium sufficit, quando non; ideo Gravitatis insensibilis formidabile nomen commentus est egregius hic vocum non rerum novarum artifex.   

§. IV. Septem absurda praeter supra refutatos errores in praedictis Dialogorum Philosophicorum libris, notantur.  
PRaeter duorum librorum Artis novae & magnae ineluctabile fatum, praedicti theorematis ruinâ, labuntur duo libri de instrumentis Hydragogicis, tum quoad theoriam, tum quoad praxes circa non ens, (Gravitatem scilicet insensibilem) plane chimaericas: quos igitur absque ulteriori examine missos sacio, quibusdam tantummodo absurdis ibidem obiter notaris.  
Primum absurdum committit Author sibimet contradicendo, dum lib. 1. de instr. hydrarg. Dial. 2. sect. 3. haec verba habet: Tubò situm horizontalem habente; ut ABC, totum mercurij pondus interioribus tubi lateribus innititur: proindeque nullum potest habere conatum exeundi apertô orificiô A, — quare nequit hydrargyrus illius tubi, utrumlibet extremorum A vel O urgere, id est, horizontaliter moveri, sed deorsum solum idque juxta lineas à terrae centro rectas ductas. Haec inquam contradicunt tum primae Archimedis positioni, tum etiam Authoris theoremati, sect. 7. dial. 1. lib. 1. dial. Philos. viz. Corporafluida, utiaër, aqua & hydrargyrus quaquaversum, uniformiter, & ex omni parte aequaliter urgent & premunt. Etiam experientiae; nam apertô utroque tubi orificiô, effluet hydrargyrus; quod fieri non posset, nisi utrumque ejus extremum mercurius urgeret.  
Secundum absurdum est quod occurrit in fine sect. 10. Dial. 1. lib. 2. de instr. hydrarg. Non improbabile hinc deducimus argumentum ad probandum maris summum, montium vertices altitudine adaequare, ubi aquarum fontes reperiuntur. Videtur auctor altissimae rupis Forthdnae Bass dictae (ut nihil de aliis loquar) perquàm oblitus, dum haec scriberet, in cujus summo vertice plusquam 50. ulnis supra maris summum, Fons aquae ducis habetur. Simile est illud problema, lib. 2. Dial. Philos. 1. sect. 12. quô docetur, ope Baroscopij investigare montiúmne cacumina, an maris summum sit altius, aequalémne habeant altitudinem.  
Tertium absurdum, est problema illud, lib. 2. dial. 1. Philos. quô ex diversis hydrargyri in Baroscopio altitudinibus, & altitudine montis aut pyramidis, per regulam Trium, Atmosphaerae altitudinem esse 6876. passuum, totiusue aëris quantitateni secundum reliquas ejus dimensiones colligit & determinat; nam secundum hanc praxin evidens est, aërem ab imo ad Atmosphaerae summum, aequaliter esse densum & gravem; aliàs proportio non tenebit: Et tamen lib. 6. Dial. Philos. 2. sect. 7. ascribit auctor aëri claterium, item lib. 1. Dial. Philos. 3. sect. 9 asserit partes aëris inferiores multò majori compressioni subesse, quam superiores; & per consequens, quò terrae propiores, eo compressiores.  
Quartum absurdum est in sect. 1. Dial. 2. ejusdem libri, viz. determinatio morae cometarum supra horizontem absque consideratione declinationis, per solam à terra distantiam; quasi ratione solius distantiae majoris, eo majorem haberet cometa moram supra horizontem, & ratione solius distantiae minoris, eo minorem: cum cometa intra octingenta milliaria ad terram, absque ullo occasu, motu primi mobilis, circumvolvi possit, (etiam supposito terrae ambitu, quem ponit Author 21600. milliarium) nempe si existat in axe mundi, vel eò circiter: & 1. contra, è stellis fixis quaedam sunt, quae nullam omnino habent moram supra horizontem, & aliae quae non ultra unam, duas, vel tres horas a nobis conspiciuntur, etiamsi lunâ sint multo altiores.  
Authoris imbccillitatem miseratus, regulam sequentem, quâ suorum cometarum supra horizontem sensibilem moras computet, construxi. 

Sinus altitud. pol. bor. = a  
Sinus altit. aequat. = b  
Cos. declin. verae = c  
Ejusdem sin. = d  
Semidiamet. terrae = e  
Distant. Cometae = f  
Sinus totus = r    
Reg. Si declinatio sit australis, erit semper cosinus arcus semidiurn● supra horizontem sensibilem ; At si haec radio major sit, nunquam oritur cometa.  
Si vero declinatio fuerit borealis, & , erit ejusdem arcus semidiurni cosinus : Et si haec radio major sit; non oritur cometa.  
At si declinatio sit borealis, & ; arcus semidiurnus supra horizontem sensibilem superabit quadrantem, eritque sinus excessus, quo dictus arcus quadrantem superat ; Et si haec radio major sit, cometa non occidit.  
Et tandem, si declinatio sit borealis, & , erit dictus arous praecise quadrans.  
Quintum absurdum, est assertio quam ibidem habet, viz. Nos juxta terrae superficiem ad 70. vel 80. milliaria prospicere posse: Cum cuilibet in Elementis, Euclidis versato, ex Prop. 36. lib. 3. & prędicto terrae ambitu, demonstratu facillimum sit, hominis octo pedibus alti prospectum ad quatuor milliaria non extendi.  
Ut ad unguem solvere possit hoc problema vel ipse Dromo, hanc a me habeat regulam generalem.  
Sit terrae diamet. = a  
Hominis altit. = b  
Srit semidiam, horiz. sensib. = .  
Hinc positô terrae ambitu milliarium 21600. & hominis altitudine 8. ped. Erit semidiameter horizontis sensibilis, milliar. 3'32. quam proxime.  
Sextum absurdum, est contradictio inter Authoris scripta; nam scribens de lagenae descensu in mare profundum, lib. 2. Dial. Philos. 4. sect. 10. sic ait, Necesse est, quum quò altior est aqua, eò validius & fortius evadit ejus elaterium: seu, quod idem est, pressura. FRANC. Fortiúsne similiter evadit incarcerati aëris elaterium? ALEX. Haud dubiè: semper tamen manet aquae elaterio debilius. Et Dial. 5. sect. 6. ejusd. lib. istis apertè contradicit; ubi enim de quodam Baroscopij intra campanam urinatoriam aquae immersam phaenomeno, & accuratissimo inter aquam ambientem, incarceratumue, aërem, aequipondio, verba fecisset, addit: Verbô, incarceratus campanae aër, eâ 34. pedibus demersa, eidemmet subest elaterij gradui, cui aqua proxima, etc.  
Septimum abfurdum, idque peccans adversus elementa Geometrica, tale est, lib. 4. Dial. Philos. 1. sect. 8. circa sinem, dum ostendit duo plana aenea rotunda diametri 3. digit. pondere praecisè 100 librarum, pati separationem, quia cylindrus aëreus, cujus pressura uniuntur, 100 libris est gravis; addit, Si duplò minoris forent diametri, tum 50. pondo sufficerent: Si duplò majoris forent diametri priore mensura, non minùs 200. libris est appendendum. Hic evidenter supponit noster Author, circulos inter se esse in simplici proportione diametrorum: quod falsissimum est: nam ex Prop. 2. lib. 12. Elem. Eucl. Circuli inter se sunt, quemadmodum quadrata à diametris, & ex Prop. 20. lib. 6. El. Polygona similia duplicatam habent: eam inter se rationem, quam latus homologum ad latus homologum: Ergo circuli sunt in duplicata ratione diametrorum: Consequenter, si corpus aeneum planum & rotundum diametri 3. digit. ad seipsum à simili separandum, pondus praecisè 100 librarum requirat, tunc planum aec●●e crassum duplò minoris diametri, requi●●t tantùm 25. libras; & planum duplò majoris diametri requiret 400. libras.  
Similiter errat lib. 2. Dial. Philos. 3. sect. 9 dum ait, Cylindrum aereum trium ●●guorum in diametro, 100 libris esse gravem-Volam manus totidem supportare. Tergum hominis proni, sextuplum. Nam sic facit volam (novem ad minimum digitos quadratos continentem) circulo trium digitorum in diametro aequalem: Et quod ad tergum hominis proni, pedi quadrato seu 144. digi●is quadratis ad minimum aequale, attinet; imo ex suppositione, quod quadratum trium digitorum, 100 tantum aërus libras supportet, (quot ipse circulus aequalis diametri posse, locis citatis affirmatur) secundum duplicatam laterum rationem, 1600. libras sustentabit; & tamen juxta Authoris nostri praxin Geometricam, 600. tantummodo libras supportat.    

CAP. II. Reliqui Dialogorum Philosophicorum libri leviter perstringuntur.  
LIb. 4. de Vacuitate prolixè satis tractatur, de quo lectorem Philosophicum appello, siquid in eo, de vacui existentia, praesertim in Baroscopio post hydrargyri delapsum, positiuè nedum solidè determinatum, indial. 2. speciatim, aut alibi reperiat: Anne sibimet hac de re contradixerit Author, affirmando in hujus libri dial. 3. sect. 8. Aliquod corpus spatium hydrargyri delapsi in Baroscopio occupare: & dicendo, sect. 3. Dial. 2. ejusd. lib. Satis improbabile esse supremam tubi partem aethere repleri; Cum nihil per Aethera intelligat Cartesius, quam corpus aëre subtilius. Númne etiam rectè assignaverit Author vulgarem opinionem, scil. Naturam penitus ab inanitate abhorrere, pro fundamento sententiae aethera astruentis; cum satis constet, talis corporis necessitatem, ex sententia Cartesij, ab identitate Corporis & Spatij unicè dependere; nequaquam verò ab ulla naturae Exhorrescentia aut Appetitu, quem solis Viventibus attribuit.  
Adhaec, lectorem Philosophicum appello, de rationibus, lib. 2. dial. Philos. 2. pro Vacuo disseminato: quantum iis insit ponderis. Num etiam Deusingio, spatiola inter partes aeris disseminata probanti esse aliquid, propter ipsorum trinam dimensionem, repugnantiam sapiant distinctiones, lib. 4. dial. Philos. 3. sect. 2. adhibitae, scil. Dimensionum, in Reales & Spatiales: Alicujus, in Aliquid reale & Spatiale; Nihili, in nihil reale & Spatiale? Númne hinc etiam sequatur, (quod alibi asseritur) Spatium esse nihil; Corpusue esse in spatio, idem esse, ac Corpus esse in Nihilo?  
Librum quintum de Antliae Phaenomenis missum facio: Libri etiam sexti tentamina ad Motum perpetuum puerilia; Libros item de Instrumentis Hydrag●gicis, quos cap. 1. §. 1. & 2. funditus eversos cuivis est in propatulo.  
Sequitur liber de Hygroscopio & Chronoscopio; de illo, praeter vocabulum, alia nonnulla ab ipso, alium quendam mutuatum esse veretur Author. Et tamen rem ipsam a Baptista Porta, si non etiam ab aliis, se habere fatetur. Ea quae de Chronoscopio, Capite sequenti castigabuntur.   

CAP. III. Probatur universa Authoris doctrina de Pendulo esse falsa.  
DIal. 3. sect. 1. de Pendulo, haec habet verba: Fateor hoc opus eò difsicilius & laboriosius fore nobis, quod neminem adhuc viderimus, quorum dicta vel scripta consuleremus, & quorum vestigijs, si opus foret, insisteremus. Ideo sect. 3. Nostrum appellat Chronoscopium, licet non recens excogiratum. Authori credo, artificium enim sapit artificem; nam ne vel unam veritatem de Pendulo demonstratam continet, sed merits est elrorum fasciculus; ut in progressu patebit. Interim, quam maxime observanda est Authoris nostri fiducia inanis, qui, absque ope Geometrica, Motus phaenomena aggreditur. Sed ad rem ipsam redeamus.  
Normam ex aere vel ferro, multo plus amplitudinis quam crassitudinis parari jubet: in cujus altero extremo fiat foraminulum, per quod ingressus claviculus suspensum radium sustentet, ut videre est in figura Sect. 2. Vult etiam normam esse 60. digitis longam, & totidem uncijs gravem. Pag. 555. Dein praecognita quaedam tradit, unde conclusionem quandam deducit, & hinc propositiones suas de Penduli Phaenomenis.  

§. I. Praecognita ad propositiones de pendulo, examini subjiciuntur.  
DIal. 4. Sect. 3. distinguit Author in pendulo, motum perpendicularem & circularem, & rursus in hoc, sect. 7. motum perpendicularem & horizontalem. Dividit etiam gravitatem in perpendicularem & circularem: Ratione illius (inquit) placidè quiescit pendulum, finitis vibrationibus in perpendiculo AB, cum appetitu tamen naturali ●endendi deorsum sublatô claviculo centrali. (vide Authoris figuram) Circularem gravitatem subdividit in circularem descendentem, & ascendentem; virtute prioris, ait pendulum ferri deorsum in semicirculo à puncto H ad B: virtute posterioris sursum ferri à B ad R.  
Quod ad distinctiones has, lectorem advertere velim, quomodo ulla gravitas dicatur ascendens, cum ipse Author, lib. 1. dial. Phil. 1. fect. 6. sic definiat, Gravitas est potentia intrinseca, quâ aptum natum est corpus ferre deorsum. Insuper, si pro qualibet motus determinatione, varias gravitatis species pro demonstrationum basi essingere liceat, novas in infinitum excogitabit quilibet, v. g. Gravitatem Horizontalem, Spiralem, Hyperbolicam, Parabolicam, Ellipticam, gravitatem Cissoidalem, etc. nam per istarum figurarum tubos potest aqua deorsum sursúmve ferri.  
En quas ridiculas comminiscitur Gravitatis distinctiones, pro qualibet motus determinatione, de qua dicturus erat: figmentis eum uti necesse est, qui ad proprietates motus explicandas se accingit, solidis, praesertim Geometricis principijs nudus: Post haec, in ordine ad propositiones sequentes, quaedam Scienda, partim absurda, partim sibi repugnantia, pręmittit; quorum  
Primum habetur Dial. 4. sect. 4. Radium scil. aeneum AB placidè quiescentem, habere solummodo gravitatem perpendicularem.  
Secundum est, Radium AB ad H usque elevatum, tum gravitatem perpendicularem, tum circularem habere: priorem, quia extractô claviculo A, positòque planô ad terrae centrum inclinante, super quod descenderet, eò indubie progrederetur, quemadmodum lapis de tecto aedificij devolveretur, semel demissus. Posteriorem habet, quia sublatô digito radium in H supportante, confestim ad perpendiculum AB deorsum ruit.  
Tertium, Radium AB ad summam altitudinem G elevatum, omnem suam gravitatem perpendicularem amisisse, atque sic solam circularem habere. Rationem prioris hanc assignat, Quia nullum habet appetitum radius movendi se horizontaliter. Ridiculum & falsum est, dicere radium AG non habere appetitum rectâ movendi deorsum, quia non habet appetirum movendi horizontaliter: Hinc enim sequeretur, (contra hujus sectionis positionem primam) radium AB placidè quiescentem, nullam habere gravitatem perpendicularem; nam non habet appetitum se movendi horizontaliter.  
Insuper ibidem scribit, Dimidium gravitatis radij ejusdem ad summam altitudinem AGNOSTUS elevati, claviculô suspendi: Hinc infero, Extractô claviculo, positoue plano ad terrae centrum inclinante, super quod descenderet, eo indubie progrederetur quemadmodum lapis de tecto aedificij devolveretur, semel demissus: (alias claviculus nullam supportaret gravitatem) Ergo secundum ipsum authorem in Sciendo secundo huj. Sect. Radius AG habet gravitatem perpendicularem: quod negavit author in Sciendo tertio.  
A praecedentibus Sciendis sibi invicem contradicentibus, infert Author Sect. 5. conclusionem hanc: Radium aneum, quò altius elevatur, eò magis gravitatem lucrari circularem: atque ex consequenti, eò magis amittere gravitatem perpendicularem: & è contrario, quò magis deprimitur, eò magis gravitatem circularem amittere, & ex consequenti eò magis gravitatem perpendicularem lucrari. Quomodo hoc probatum sit, Authoris verba (etiam admissa ejus distinctione gravitatis fictitia) manifestabunt: Inter probandum, hac etiam utitur ratione. Ideo nequit pendulum AF (in fig. pag. 555.) plus gravitatis perpendicularis, quantùm ad motum rectà deorsum, habere, quam sunt unciae Radij inter 2 & A, & ratio est, quomam meo digito ejus extremum F: supportanti tot radij unciae innituntur, quot sunt digiti inter 2 & C, qui sunt propemodum quatuor, & ex consequenti radius sic elevatus minùs gravat claviculum, quatuor uncijs, quam radius perpendicularis AC; Quid multis? Claviculus supportat uncias radij AF: quinquaginta sex, digitus verò quatuor.  
Hinc contra authorem infero. Ergo digitus similiter supportabit 60. uncias radij AD, (quia tot sunt digiti inter A & C) & claviculus supportabit nullas, (quia totus radius ex suppositione continet tantum 60) quod est gravissimum absurdum: & etiam contradictorium Sciendo 4. sect. 4. Item isti quod habetur Dial. 5. sect. 1. lin. 9 Quarto, claviculum 30. solum uncias penduli, ad summam altitudinem AD elevati sustinere; & digitum, cui alterum radij extremum D innititur, totidem supportare, summatim 60.   

§. II. Propositio prima de Pendulo ostenditur ridicula esse, vel falsa.  
EX praedictis Dial. 4. sect. 6. propositionem hanc primam demonstrare conatur; viz. Progressum diminutionis vibrationum penduli, esse juxta sinuum proportionem, id est, singulas vibrationes alternatim se invicem breviores esse, eadem proportione, quâ inaequales divisiones semidiametri AC, sunt se invicem ampliores. Et Sect. 7. Hinc clarissime ostenditur quomodo penduli vibrationes sunt proportionales ad sinus, nam posito, quod à summa altitudine demissum, ad S usque vibraret, oportet provehatur horizontaliter inter N & MR, in primâ vibratione. In secundâ ex MR ad O. In tertia ex O ad P. In quarta ex P ad Q, & ita deinceps: sed illarum divisionum decrementum est ipsorum sinuum, ut patet, conferendo eas cum semidiametri divisionibus AC. Et addit: Penduli vibrationes diminui cum proportione ad sinus, quatenus ejus motus est Hortzontalis; non autem quatenus est perpendicularts, aliàs forent conformes etiam inaequalibus divisionibus semicirculi BCD, cui experientia, teste oculo, contradicit.  
Vel hic intelligit author (dum de proportione sinuum loquitur) relationem quam habent sinus ar cuum aequidifferentium, ut apparet ex ejus figura; vel nihil solidi: cum sinus omnem inter se habere possint proportionem: & si relationem intelligat jam dictam, erunt arcus decrementorum omnium vibrationum inter se oequales, quod ipse fatetur experientiae contradicere: quodue absurdum ipse secuturum infert, si vibrationes dicerentur diminui, cum proportione ad sinus, quatenus ejus motus est perpendicularis; cum tamen per easdem vibrationes consideratas ut motus Horizontales, describantur sinus recti, & per easdem consideratas ut motus perpendiculares, describantur sinus versi: & per easdem prout considerantur distantes à linea horizontali, describantur cosinus arcuum vibrationum.   

§. III. Propositio secunda de Pendulo rejicitur.  
PRaecedenti propositioni falsae aut ridiculae confisus, hanc secundam demonstrare tentat. Scil, Omnes vibrationes penduli esse Synchronicas. Quae quoniam priori hactenus eversae innititur, cum caeteris pariter est rejicienda; observatô obiter unico petitionis principij levi vitio, quod in ejus demonstratione, pro more solito, committitur.  
Arguit enim author ab incremento gravitatis circularis descendentis ad incrementum velocitatis: quam Gravitatem, ibidem sciendum ait reipsa idem esse, nempe respectu penduli motus, cum velocitate: & hinc est, ut quot uncias gravitatis acquirit pendulum ex E ad K elevatum, tot revera gradus velocitatis acquirantur; quibus penduli motus efficitur velocior. Hoc est, (per authoris Sciendum jam dictum) quot uncias velocitatis acquirit pendulum, tot revera gradus velocitatis acquiruntur.   

§. IV. Rejicitur reliqua authoris doctrina Dialogo 5. tradita, de hactenus dictis, & caeteris penduli Phaenomenis.  
DIal. 5. reliqua Chronoscopij Phaenomena proponit explicanda: ubi Sect. 1. modum computandi incrementum gravitatis penduli, inter ascendendum, primum aggreditur. Ubi notandum est, quod Dial. 4. sect. 6. lin. 14. ad probandum progressum diminutionis vibrationum penduli esse juxta sinuum proportionem, hoc medio usus fuerat: Quia câdem proportione diminuitur radij gravitas ex C ad K; vel L vibrantis, quâ inaequales divisiones semidiametri CA evadunt se mutuò ampliores. Et ibidem dixerat, incrementum illud gravitatis, quod acquirit pendulum inter ascendendum, esse proportionale ad sinus. Haec, ut dicebam, ad praxin reducere conatur, Sect. 1. docendo methodum supputandi numerum unciarum radij aenei penduli 60. uncijs gravis, quas supportat claviculus centralis, & quas digitus, pro singulis penduli elevationibus. Haec sunt ejus verba. Sed quomodo definitè nôsti claviculum supportare 35½ uncias penduli AM, & uncias 40. penduli ALL? (Vide fig. pag. 564.) ALEX. Extende circini mucrones inter 8 & C, & sumptô hujus distantiae dimidio, applicetur alterum circini extremum puncto M, atque oppositum in puncto N terminari invenies. Docet hoc, claviculum tantò plus de gravitate penduli AM sustinere, quam penduli AD, quantò distantia AN est major AX, quae est digitorum 5½.  
Quomodo haec cohaereant, judicet lector: locis citatis, indefinitè loquitur de incremento, & etiam de decremento gravitatis Penduli inter ascendendum. Sibi quoque adversatur, nam locô prius citato, dicit incrementum gravitatis inter ascendendum esse proportionale ad sinus; & tamen illud per dimidia sinuum versorum hic loci supputat. Sed si veri penduli, h. e. globi filo appensi gravitatem pro quavis elevatione congruè computare velit author, hac regula sequenti utatur. 

Si sit penduli longitudo = r  
Gravitas globi dum in linea perpendiculari quiescit = b  
Sinus elevationis penduli = a  
Erit globi gravitas in elevatione data =    
Hanc gravitatis computandę methodum sequuntur praedicta & etiam reliqua penduli Phaenomena ab Authore demonstranda. Scil. sect. 4. Penduli vibrationes juxta sinuum proportionem diminui. Sect. 5. Eas esse Synchronicas. Sect. 6. Incrementum velocitatis penduli inter descendendum esse ad sinus proportionale. Sect. 7. Pendulum tam citò quadrantem circuli percurrere, quam corpus ejusdem gravitatis & figurae semidiametrum. Sect. 8. Incrementum velocitatis penduli esse non tantùm proportionale ad sinus, verùmetiam esse juxta ordinem numerorum quadratorum, ab unitate initorum, in spatijs post aequalia tempora confectis. Sequentibus sectionibus, adducit argumentum Riccioli, quasi suum, ab incremento velocitatis corporum descendentium, adversus Copernici sententiam de motu Telluris, tanquam invictissimum.  
Quod ad primum, pari efficacitate id probat, quâ anteà: atque insuper hic loci novae & falsae nititur Hypothesi, nimirum, Quod nullae aliae possint excogitari divisiones, quibus proportionales dici possunt vibrationes, quam arcus & sinus; Cum tamen omnes lineae possunt infinitis diversis rationibus in partes inaequales dividi.  
Demonstratio secundi & tertij phaenomeni, novae & falsae nititur hypothesi: viz. Phaenomeno Sectionis septimae: quam, praeterquam quod quivis experientiae adversari comperiat; falsam esse, ex duobus postulatis sequentibus hic demonstrabitur.  
Postul. 1. Duos globos ejusdem ponderis & magnitudinis, integram diametrum perpendicu larem AB circuli ADBC, & quamvis ejusdem circuli chordam diametro perpendiculari conterminam, aequali tempore percurrere. 
Fig. 2.   
Hoc extra omnem contraversiam est positum, & à Galileo notatum, System. Cosm. dial. 4. pag. 335. secundum impressionem Lugdunensem.  
Postul. 2. Omnes ejusdem penduli vibrationes esse Synchronicas. Hoc est ipsius Authoris.  
Hinc contra Authorem demonstraturus sum, Duos globos ejusdem magnitudinis & gravitatis, seu (quod idem est) eundem, circuli semidiametrum GB, & quadrantem DEB, aequali tempore non percurrere. Sumatur arcus EB indefinitè parvus, ita ut non differat à sua chorda EN, faciendo differentiam omni quantitate assignabili minorem: Ergo, cum (per Pôstul. 2.) globus idem quadrantem DEB, & arcum EB, aequali tempore percurrat: aequali etiam tempore percurret quadrantem DEB, & chordam EN: sed aequali tempore percurrit chordam EN, & Diametrum AB, per Postul. 1. Ergo, aequali tempore percurret quadrantem DEB, & diametrum AB; sed (juxta hunc Authorem) aequali tempore percurrit quadrantem DEB, & semidiametrum GB; Ergo, aequali tempore percurrit integrum diametrum AB, & semidiametrum GB. Quod est absurdum. Ergo, globus non percurrit circuli quadrantem, & semidiametrum, aequali tempore. (contra quam volebat hic author) Quod erat dem.  
Demonstratio quinti phaenomeni, viz. Incrementum velocitatis penduli esse juxta ordinem numerorum quadratorum, est, ut reliquae, parenti similis; oftendit enim mirabilem centralis claviculi influxum in penduli motum, pro singulis momentis ad finem usque; ejusque efficientiam in penduli velocitatem cum proportione ad sinus: at hoc leve! Innititur praeterea haec dicta demonstratio Phaenomeno quarto, quod falsum esse jam demonstravimus.  
Praeter errores supra refutatos, authoris nostri ignorantiam phaenomeni istius in pendulo, quod jampridem omnibus tritum est ac vulgare, ob nimiam ejus jactantiam & insolentiam, absque nota praetereundam non esse censeo. Phaenomenon est hoc.  
Si sunt duo gravia aequalia & similia, B & D, filis AB, & CD appensa: Tempus vibrationis penduli AB est ad tempus vibrationis penduli CD, in subduplicata 
Fig. 3.  ratione AB ad CD; seu in ratione AB ad G mediam inter AB & CD Proportionalem. Quod in gratiam authoris nostri sic demonstratur.  
Sint AB = A, CD = CF, Tempus vibrationis penduli AB = M, Tempus vibrationis penduli CD = N. M est tempus quô grave B cadit ab E, & N est tempus quô grave D vel idem B cadit ab F: & ideo, , . Ergo, . Quod erat dem.  
Hinc in gratiam authoris, hanc etiam regulam construxi. 

Vnius penduli longitudo sit = a  
Alterius longitudo = b  
Prioris tempus vibrationis = c  
Erit alterius tempus vibrationis =    
Proprietatem hanc penduli, quod nostrum appellat, eum penitus latuisse, ex dial. 6. de Chronosc. sect. 12. omnibus conspicuum est. Si unquam audiverit, ratio cur eam scriptis suis non inseruerit, facile assignari potest haec; proportionum ignarus subduplicatam rationem non intellexit: quod ex scriptis ejus praesertim Hydrostaticis, ubi proportionem Directam & Reciprocam ubique confundit, clare cernitur.  
Dialogum quintum claudit argumento, contra Copernici sententiam, ab incremento motus gravium desumpto; de quo quasi invictissimo Thrasonem agit; & licèt primus omnium eo usus fuerit Ricciolus, ejus tamen nulla facta hic mentio. Dicitur hic, Necessariam esse connexionem inter motum terrae vertiginosum, & incrementum velocitatis descendentium apparens solum: Quod incumbit probandum. Asseritur item, Copernicanos ad unum omnes, incrementum reale velocitatis negare: Quod falsissimum est. Quid ponderis huic argumento insit, extra omnem contraversiam, adversus Ricciolum non ita pridem posuere Stephanus de Angelis, & Andreas Tacquet, uterque licèt Pontificius: quorum rationibus tandem ille succumbere coactus est, ut manifestum est ex Transact. Philos. pag. 870. & alibi; quare actum agere supersedeo.  
Nihilominus authorem monitum volo, argumentum hoc falsa suffulciri hypothesi, scil. Lineam curvam in qua descendit grave cadens, esse circularem: quam praedictus Stephanus quandam esse Spiralem demonstrat, cujus proprietas est haec. Quod rectae (in Riccioli & authoris figura pag. 578.) sumptae ad libitum, HQ, IR, semper sunt in duplicata ratione angulorum HAD, JAD. Et nunquam ad Circulum appropinquat, nisi grave ad terrae centrum spatiô sex horarum decidat, quod in casu Riccioli & authoris nostri fit spatio 21. 53. Imo datâ at non concessâ Riccioli suppositione, quod praedicta linea sit circularis, nullatenus tamen inde tollitur incrementum velocitatis reale: Quod si hac de re dubitare pergat noster author, primo rogatu satisfaciet è Pedellis, alter.  
Ego intereà, ne caeteris magnis quidem illis artis revera parvae immorando nugis, nimia lectori creetur nausea, ad examen Tyrociniorum Mathematicorum vcrbô expediendum memet accingo: in quo, ut ex cauda catum dignoscat lector, sufficiat sequentes annotasse errores.     

TYROCINIORUM MATHEMATICORUM EXAMEN.  
DIcit itaque (1) noster Tyro (modò hoc sit insigniendus nomine, quem ne vel prima Matheseos elementa primoribus degustasse labris certo certius est) pag. 26. Horas planetarias distingui per circulos; quas per lineas mixtas fieri norunt Gnomonici omnes.  
(2.) Asserit pag. 50. Sub circulis polaribus, Gnomonum extremitates in horologijs horizontalibus, ut semel ab Aequatore digressus est Sol, Parabolas describere: Cum tamen in horologio horizontali describatur Parabola, solummodo dum Sol est in Tropico proximo: Et extra hunc (nisi in Aequatore) semper describantur Hyperbolae.  
(3. Pag. 52. dicit, Gnomones & stylos suis extremitatibus describere Ellipses, in zona frigida. Quod verum tantummodo est, cum Sol non occidit; nam cum occidit, semper describitur Hyperbola, nisi in Aequatore: Et cum mediâ nocte horizontem radit, semper describitur Parabola.  
Ne amplius noster hic Tyro, sub Polaribus aut terrarum alibi, in sectionibus Conicis sciaterico horizontali, aut cuivis alij inscribendis erret: has regulas generales observet.  
Reg. 1. Vbique terrarum, quando Sol occidit, describitur semper Hyperbola in plano horizontali, nisi Sol fuerit in Aequatore, & tunc describitur linea recta.  
Reg. 2. Quando Sol non occidit, semper describitur Ellipsis; nisi idem fuerit horizon cum Aequatore, & tunc describitur Circulus.  
Reg. 3. Quando Sol horizontem lambit, describitur Parabola.  
Not. Quod hic dicitur de horizontali, de quovis alio plano super quod Sol occidit, non occidit, aut tantùm lambit, intelligendum esse.  
(4. Pag. 100 Dum distantiam duorum locorum, quorum alter sub aequatore sit positus, inquirit; Proportionis terminos sic statuit. Vt est radius totus ad complementum differentiae longitudinis; ita complementum latitudinis datae ad complementum distantiae quaesitae. Egregiè hallucinatur tum in vocabulis artis, nam, non radius totus, sed sinus totus, vel simpliciter radius dicere debuit: tum in ipsa arte, nam proportio sic se habet. Vt radius, ad sinum complementi differentiae longitudinis; ita sinus complementi latitudinis datae, ad sinum complementi distantiae quaesitae.  
(5.) Denique ubi loquitur pag. 120. de Echo taciturna, dicit, In quolibet speculi Elliptici puncto non potest exaudiri hujusmodi echo, sed in ipso tantùm puncto concursus. (radiorum scil. soni reflexi) Sed in speculo Elliptico nullum tale punctum concursus agnoscunt Mathematici; nam in Ellipsi duo sunt foci, in quorum uno debet statui corpus sonorum, & in altero auris audientis.  
Quâ fronte, Methodum suam Echometricam, in praefatione, Geometricam designarit author, cum non nudae figurae Geometricae, verùm demonstrationes methodum Geometricam constituant, lectori dijudicandum relinquo.  
Atque jam habes, Candide & Erudite Lector, animadversiones hasce leves, in nugivenduli nostri Scioli egregia & erudita opuscula scombris & thuri jure merito aeternum consecranda; quas à me invito, Ardelionis istius insolentia, impudentia & arrogantia extorserunt: quibus virtutibus fretus & inflatus, non tantum in varios exteros, viros eruditos & celebres, rixatricis & furiosae mulierculae in morem debacchatus est; sed etiam, ut est os homini osseum, & frons plusquàm serrea, varijs suis compatriotis, nominatim Professori primario inclytae Academiae Glasguensis, quam Salgucensem 
* Art. nov. pag. 296.  vocat, (viro, quem norunt omnes summô animi candore, vitae integritate, multâ & omnigenâ eruditione, praecipuè verò linguarum trium & omnium Orientalium peritiâ praeditum, insignem & ornatum) insolenter & impunè hucusque insultare ausus est 
* Art. nov. pag. 472.  Quare nullus dubito, quin meos conatus aequi boniue consulturus, & veniam mihi daturus sis, sicubi tibi visus fuerim paulò acerbiùs adversarium tractasse, cujus insulsa petulantia, & insignis procacitas, vel psam mansuetudinem, satyram scripsisse cogeret. Interea, ut relaxetur tibi animus ab aegritudine, aut indignatione, quam censeo non potuisse non contrahere, modò pensiculatiùs cogitaverit, quantam & qualem ignominiae notam patriae suae inurere, quem fucum & frandem literato orbi facere conatus sit famosus meus antagonista, puerilium, ridicularum, & trivialium tricarum miseram & miserè consutam farraginem sub adeo amplis & speciosis titulis praelo committendo. Interea, inquam, ut relaxetur tuus animus à praedicta aegritudine, ne dedigneris tuos oculos convertere in sequentia Tentamina Geometrica, quae sat scio, fatebere aeue virum & veram Mathesin sapere, ac quae à me ad examinis incudem modò revocata sunt, nauci hominem, supinam inscitiam, crassam & stupendam ignorantiam, tum Matheseos, tum naturalis Philosophiae, altâ & clarâ voce singulis pronunciant & proclamant: Quae denique examini subjicere, modò capiat, aut ad Graecas Calendas capere possit, nostro per me licebit adversario. Interim tu iis utere, fruere, & Valerius   

TENTAMINA QUAEDAM GEOMETRICA DE Motu Penduli & Projectorum.  
I. SInt rectae A, DB, horizonti parallelae; situe tempus (quo descendit grave in recta CD) 
Fig. 4.  , & , , : Esset tempus (quo descendit idem grave in recta AB) .  
II. Et positâ in D; foret velocitas in . Haec facile eliciuntur ex Galilaei, & aliorum de motu demonstrationibus.  
III. Sint deinde , , , ; ponitur enim AG ipsi FE perpendicularis. Descendat itaque grave per rectam AB, cujus velocitas in B sit : descendat quoque idem grave per rectas AF, FD; erit ejus velocitas in . Hoc ex antecedente nullo negotio deducitur; modo animadvertatur mobilis, quod in diversis rectis movetur, impetum seu velocitatem mutari in rectarum occursu, ita ut velocitas in prima linea sit ad velocitatem in secunda, in ratione radij ad cosinum inclinationis mutuae rectarum. Ut in figura, cum motus perficitur in diversis rectis AF, FD; velocitas, quam acquirit grave descendens in F, mutatur in aliam in FD, quae priore minor est in ratione FG, ad FA: Atque hoc verum est in omni motu, sive aequali, sive quovis modo accelerato vel retardato.  
IV. Hinc igitur colligimus motuum praescriptas velocitates variari tantum propter linearum inclinationes, in quibus diriguntur. Et proinde si nullae tales sint inclinationes, nullae etiam sunt velocitatum ab ordinatis differentiae: atque in lineis curvis nullae tales sunt inclinationes, & ideo in lineis curvis mobilia eâdem velocitate incedunt, qua in lineis rectis. Hisce in genere pensitatis, dico grave eâdem velocitate moveri, sive in linea curva, sive in recta descendat; nam eruditis hactenus innotescit, grave eâdem velocitate moveri, sive in recta horizonti perpendiculari, sive in recta eidem inclinatâ descendat. Non arduum foret, hoc in penduli descensu Geometrice demonstrare per hujus tertiam, ab exhaustione Archimedea: sed prolixior est haec summi Geometrae methodus, quam permittit instituti brevitas. Novi hanc doctrinam Galilaei experimentis non congruere, dum asfirmat mobile citius descendere per arcum circuli, quam per ejusdem chordam; & citius per duas chordas, quam per unam: Item (quod hinc emergit) breviores penduli vibrationes tardius persici, quam ejusdem longiores. Sed vereor Galilaeum deceptum esse à gravium elaterio motum praecipitante, quod hic summopere advertendum est, & seorsim considerandum. Utcunque sit, super hac hypothesi, de tempore quo perficitur penduli vibratio inquiramus.  
V. Sit igitut AHF circuli quadrans, ex hujus puncto C demittatur pendulum. Ducatur radio & horizonti AH parallela EGLANTINE, & huic perpendicularis CD: pendulum in G eandem habet velocitatem vel impetum, quam habuisset in D, si ex 
Fig. 5.  C demissum fuisset. Quaestio nunc est quam cito descendit à C in G? Sit , , , , . Erit tempus, quo pendulum descendit ex C in  
VI Altitudines penduli vibrationum, seu ipsarum sinus versi, sunt quam proxime in subduplicata ratione quantitatum harmonice continue proportionalium: atque hinc videtur sequi corporis gravis per centrum terrae vibrationes esse in eadem ratione.  
VII. Affirmant non pauci in projectorum jactu perpendiculari aequales impetus sub eadem altitudine tam ascendenti quam descendenti mobili inesse: quod mihi nequaquam arridet: cum hinc clare sequatur, motum projectorum, exclusa gravitate, esse aequabilem: & gravis vibrationes per centrum terrae omnes inter se esse aequales, atque motum hunc in perpetuum duraturum: imo ipsius penduli vibrationes aequales & perpetuae forent.  
VIII. Motus projectorum, exclusa gravitate, videtur aequaliter retardatus; nam unius medij homogenei, quale hic supponimus nostrum aërem, una semper est resistentia; quod impedimentum de novo semper adveniens motum producit aequaliter retardatum.  
IX. Propositum nunc sit inquirere, qualis sit linea à motu projectorum descripta, secundum nostram hypothesin composita ex uno motu aequaliter retardato & altero gravitatis aequaliter accelerato. Sit igitur linea recta VK, in qua perficeretur motus projecti exclusa gravitate, & 
Fig. 6.  recta AK (eidem VK perpendicularis) tempus in quo motus perficitur. Tempore AB sit projecti ob gravitatem descensus BF; ducatur nunc parabola AFI, verticem habens A, & compleatur rectangulum AKUX, fiatue parabola XYK, cujus vertex K. Sit tandem curva à motu projecti descripta VTRPL; sintue datae rectae , , , ; rectae vero indefinitae sint , : sintue , latus rectum parabolae , latus rectum parabolae . Manifestum est tempus , , ; & proinde , , & ideo , , : Ope harum trium aequationum, ablatis quantitabus r, l, x, fit . Fiat nunc , & emergetue sequens aequatio & ad tollenda signa radicalia, utramque aequationis partem in se multiplicando, & radicem quadratam extrahendo Unde innotescit curvam VTRL esse parabolam, cujus constructio est satis expedita, cum V sit ejusdem vertex, & V β ipsius diameter, factis , & rectis V α, α β, ipsis ST, SV, parallelis: hinc innotescit V β (cum detur angulus VOP) quae sit ; & proinde parabolae latus rectum est . Ex praedictis facile colligitur rectam VK tangere parabolam in V, & KL eandem tangere in L.  
X. Si detur recta , ejusque elevatio supra horizontem PUL: oportet ita elevare machinam VS, ut projectum decidat in P quoniam OP perpendicularis est ad horizontem, datur angulus OPV, cujus finus , situe anguli ignoti OVP sinus , & anguli VOP sinus , sinus totus , item ; ope harum trium aequationum, & prioris quae quantitatis g valorem exhibuit, auferantur quantitates ignotae x, c, g; & ostendet ultima aequatio restans post ablatas dictas quantitates valorem ipsius v sinus quaesiti.  
XI. Hinc quoque deducitur. Si grave ascendens perpendiculariter tempore k perficiat z, & tempore s perficiat t, tempore n perficere Hac ratione adhuc projectum perpendiculari jactu in eadem altitudine, tam ascendens quam descendens eundem habet impetum; & praeterea ex una altitudine velocius moveretur, & ex alia tardius; quae duo sunt absurda summopere evitanda: at videntur provenire potius à recepto Mathematicorum experimento, nimirum, quod gravium descensus sint in duplicata ratione temporum, quam à nostro commento de projectorum motu aequaliter retardato. Sit enim tempus AB, quo projecti ascensus fiat BY; item tempora AC, AK, AD, eorumque ascensus respectivi CG, 
Fig. 7.  KM, DO; ita ut generetur curva AGN, cujus vertex G. Satis probabile est AGN esse Geometricam quandam & uniformem curvam, cum accelerationes & retardationes gradatim & successive fiant; at multorum experientia testatur GMN esse parabolam ab axe GC, & propterea GIA esset etiam ejusdem parabolae pars altera; quod tamen non videtur rationi congruere. Nos potius existimamus AGN esse quandam hyperbolam (vel saltem hyperboliformem) ita ut AC sit minor quam CN, cujus diameter ducitur à G ad punctum medium rectae AN: hac enim ratione, GO in parvis descensibus, quales fere hucusque tantum sunt observati, parum differt à curva parabolica; at ex magnis altitudinibus, cum motus acceleratus accedit quam proximè ad aequabilem, considerabilis forte accedet dissimilitudo; tunc enim hyperbolae curva vix differens ab ejusdem asymptota recta, motum quam proximè aequabilem repraesentabit. Qualiscunque sit curva AGN, haec est una ejus proprietas: exclusa gravitate, sint temporum AB, AC, AK, AD, respectivi ascensus BH, CF, KP, DE; eritue AHFPE parabola: ductâ GQ curvam tangente in puncto G, fiant arbitrariè AB, GL aequales; erit ML aequalis rectae IH, & figura GLM aequalis figurae AIH.  
FINIS.   

Errata.  
Pag. 56. l. 12. pro sit, lege sit. p. 57 l. 2. pro altitudinem, lege altitudinum. p. 62. l. 9 pro hydrarg. lege Hydrag. p. 63. l. penult. deal 1. p. 72. l. 9 pro far, lege ferri. pag. 75. l. 4. & 9 deal: & l. 12. deal. p. 83. l. 3. pro integrum, lege integram.   

 geometric figures