WHilst those miscellanies were printing, I met with two of Master Darys friends together, at their office in Holborn and they related to me that Master Dary had a Book in the press, in the Preface whereof he had a quarrel with me, and that he was resolved to vex me; my answer was, that if he gave me bad language I would lay it under my feet; but if he gave bad mathematicks I would return it to him again; therefore, all those calumnies, that bad and scurrilous language, (for such are their on­ly demonstrations) either by him or by any of his crew in that preface given, or may be hereafter given in any of their writings, I shall take no further no­tice of, but shall ever lay them as dirt under foot; but I shall prosecute a close conviction of their er­ronious principles in Geometry. That preface those [Page 2] authors divides into two parts, the first against Stereometrical Propositions, the second in defence of the ART of practical gaging; and as they have little to say against the first, they have as little to say in defence of the latter, but in both I shall easily sub­vert their crippled arguments. To the first,

In the first page of the preface, Master Dary hath it thus; in the tail of which Book there is a whole broad side. Here he is outragious because he was so per­fectly confuted in the tail of the guide to the young gager: truly, as it was the confutation of the ART of practical gaging; it deserved no better preferment, than to be put in the tail of the young gagers guide: however, if I find Master Dary's understanding im­proved by my instructions there given, in these his Miscellanies; I shall to encourage him, commend him in the tail of this.

In the second page of the Preface, he hath these words, the word frustum pyramide I cannot under­stand, But if he had said frustum of a pyramid, &c. this complaint may consist of three parts, first, frustum pyramide; second, I cannot understand; third, frustum of a pyramide; if we compare the first with the last, we shall find them both of one signification, for frustum signifies a broken piece, therefore it is as well sense to say a broken pyramid, as to say a piece of a pyramid; one familiar example for many, a broken knife, as to say a piece of a knife. Such expressions are brief and well under­stood, both signifying the same thing, and he him­self using the same expressions in the 31 and 32. 89 pag, of the Art of practical gaging: thus, is the content of the frustum pyramid, and in 29 page of that Book of Art, you have it thus, Master Michael Dary, an inge­nious [Page 3] Artist and practised gager: when it is the fru­stum of a Cylindriod: here we find him a giving him­self a good character; and then telling us of a fru­stum of a Cylindriod, and in page 32 of that Book of Art, thus, to cut this Cylindriod, here we may ob­serve this ingenious artist, how in one page he calls it a frustum of a Cylindriod, and immediately he calls the same solid a Cylindriod, so he makes no difference betwixt the part and the whole. Further, as for those mighty words of art, to wit, Cylindro­id, prismoid and peripetasma; I shall say only this,

For his 2. complaint, that is, I cannot understand; it troubles me to hear it, yet I see his understanding mend a little, as we shall observe hereafter. In the same page, he is angry because that irregular fru­stum is cut into so many parts, if that do not please his worship he may take one of the other ways which cuts that solid into fewer parts; for there are four ways every one lesse work than other. But the gunner and his crew must be a shooting though but with pot-guns.

In the third page, our gunner gives the seventh prop. of the 5 of Diophantus a Broad-side; thus, the stress of his argument is weak and infirm. Though we should grant Z equal to ⅗, it is yet to demonstrate that Z is 3 and A 5. A by supposition was an unit, then re­duce them to one denomination, and that denomi­nation being rejected, Z will be equal to 3, and A equal to 5▪ this he looks upon as an hard demon­stration, which I am not bound to tell him how to do, saith he. Further, he dwindles to his Reader, hoping for glory and would know, Whether this pro­position [Page 4] hath any relation at all to gaging. I answer yes, and argue thus, numbers have relation to ga­ging, this Proposition is of Numbers; there­fore this Proposition hath relation to Gaging. Again, triangles have relation to gaging, this pro­position is of triangles; therefore, this proposition hath relation to gaging. This prop. which he quarrels with, is the 7. of 5. of Diophantus, as is ci­ted in the 106 page of Stereo. Prop. and seeing he hath so much immodesty as to say his arguments are weak and infirm, I shall set down the text as Bache­tus hath it. Esto primus 1 N. secundus unitatum quot­libet, puta 1. & est productus eorum multiplicatione 1 N. summa vero quadratorum est 1 Q + 1. adde 1 N. fit 1 Q + 1 N + 1 aequalis quadrato. Esto latus ejus 1 N − 2. fit quadratus 1 Q + 4 − 4 N. aequalis 1 Q + 1 N + 1. & fit 1 N. ⅗ ad positiones, Erit primus ⅗, secundus 5/5: & abjecto denominatore, erit primus 3. secundus 5. & postulatis respondent. So then I have these witnesses on my side, 1. Diophantus the author of the proposition, 2. Xylander. 3. Ba­chetus. 4. Stevin. 5. Albert Girard. Those four commentators upon Diophantus every one of them setting it as above. 6. Truth it self, and it will pre­vail. So then, if Mr. Dary cannot bring better au­thority then these on his side for the stress of his argument; I shall conclude, that pride and igno­rance is baffled; and where he saith I fling dirt in the face of Van Schooten, I may very well say he flingeth dirt in the face of these 5 authors, yea and in the face of truth it self. Further, had this 7. and 8. prop. of the 5 of Diop been observed by the propo­ser and resolver of that question, it is very likely it would not have been proposed by the one, nor re­solved [Page 5] by the other; however, what I said concer­ning Van-Schooten and des Cartes is true and just, therefore no dirt. In the fourth page, our gun­ner hath more fire-works, to wit, his note for progres­sions is invalid and of no force. For saith he, there is no need of unity for the first terme of this progression. My answer is, that note for progression is of force and truth, and unity of use; thus, the question it self requires whole numbers; the seventh of the fifth of Dioph. finds whole numbers; therefore grea­than an unit; therefore well limited.

The second part, in defence of the Art of Practi­call gaging, and it begins in the fifth page, and there he telleth his Reader how he hath been com­mended by divers artists in this City. Here he ap­peals to men as ignorant, as himself is vain glori­ous. In the 6 pag. he flingeth dirt in the face of the printer, thus, in which I see there are many press-faults; that is false, they are the segment makers faults, for the segments are the complement of one to the o­ther to 100000 &c. therefore no printers faults. In the sixth and seventh pages, he sheweth how to calculate a table of segments, and here his under­standing mends a little, for he works pretty well since the last time I taught him; so then, as one mends in his Rules, so I hope the other will mend in his calculation, (with that instruction I formerly gave him) so we may expect a better table of seg­ments some time or other. In the 8. page he again dwindles and would fain insinuate into the affection of his Reader; and make him believe that I did not know that there was a third &c. differences in the table of segments, to speak the truth, that table of segments was calculated so falsly that the first diffe­rences [Page 6] did manifestly shew it; further, If Mr. Dary had known that way, or any other way better to examine Tables by, before he published those se­gments; more shame to him to publish such false ta­bles, without examination. In the 9 and 10. pa. the Gunner has fire and gun-powder, viz. know ye not that the Table for wine, ale and beer, are capable but only of the first and second differences. If so, more shame to the Calculator that they have more diff. and they so much confusedly put. As for that Book entituled A guide to the young gager, I knew not the man nor heard of the Book untill a great part of it was printed, neither did I see one line of that part of it, till it was publickly exposed to sale. Thus have I passed through this fiery conflict, and have not heard the bounce of one gun, nor receiv­ed any harme, which makes me conclude our gunner and his crew are as bad marks men, as they are segment makers, for he promi [...]ed at the begin­ning of his preface to charge his guns and pepper me. Thus have I considered him as a Gunner with his Crew; now will I consider him as a Geometer, with his famous Companions.

IN the first page of the Preface, he saith, Most whereof have lain by me many years: If so, I hope very true.

1 In the second page of the Preface, saith he, For although the sides thereof be continued, they would never be included or terminated in one point, as the Pyramide is; that is, the sides of a Pyramide are included in one point, which I deny, thus; a point hath no part, by 1 def. 1 Euclid. A Superfices (for such are the sides of a Pyramide) have length and breadth 5 def. 1 Euclid. That which hath no part, to include that which hath length and breadth, is absurd; that's a lumping point for an able Any­list.

2 In the fourteenth page, saith he, The 3 Angles of any Spherical Triangle being given, there are like­wise three sides of another Spherical Triangle given, whose Angles are equal to the sides of the former Tri­angle. Here the Gentlemen forgot to complement, and I presume in the next they will forget all good manners. Further, the sum of the sides of any sphe­rical [Page 8] triangle, are less then two semi-circles, Reg. 39▪ of 3. The sum of the three angles of any sphe­rical triangle, are greater then two right angles, but less then six, Reg. 49 of 3. therefore the Rule is false, except the sum of the three sides be grea­ter then two right angles; but the Rule is set down general, therefore a general error.

3 In page 21. we have it thus; If a sphere be by a plain touch'd, and the eye be placed at the center of the sphere, then a right line infinitely extended from the eye to any assigned point in the spherical surface, shall project the assigned point upon the plain. Here the Radius of the Sphere is taken to be infinite, for, saith he, then a right line infinitely extended from the eye to any assigned point in the spherical sur­face; but the plain is without the sphere, there­fore beyond infiniteness it self, which is absurd: however this proves them to be infinite Projectors.

4 In page 29. at the 18th it is thus; If a sphere be inclosed in a cylinder, and that cylinder be cut with plains parallel to its base, then the intercepted rings of the cylinder are equal to the intercepted surfaces of the respective segments of the sphere; that is false: For▪ Hemisphaerii superficies aequalis est superficiei cur­vae cylindri eadem ipsi basim, & eadem altitudinem habentis, saith Torricellius at the 18. Prop. de sphae­ra, & solidis sphaeralibus lib. prim. and as the whole, so the parts, by the 19. Prop. of the same. Here we have a combat betwixt Torricellius and our Geometers: First, they say the intercepted rings of the cylinder are equal to the intercepted surfaces of the respective segments of the sphere. Torricellius proves, that the intercepted superfices of the cylinder, are equal to the intercepted super­fices [Page 9] of the respective segments of the sphere. 2. These Geometers say, if a sphere be inclosed in a cylinder, here we may make the Diameters of the base of the cylinder of any magnitude, grea­ter then the diameter of the sphere, and yet the sphere be inclosed▪ Torricellius proves, that the cy­linder and hemisphere must have the same base. 3. These Geometers regard it not, whether the sphere and cylinder are upright or inclining. Tor­ricellius by construction makes them upright. Thus do these Geometers make solid superfices, for a ring is solid.

5 In page 33. they set down a rule for the sphere, and conclude it will hold in the spheroid; this rule will also hold if it were the Frustum of a Spheroid, putting d [...]d equal to the fact of the right angled conjugates in the base. That is false, by 21 of 1 of Apollonius, and 31 and 33 of Archimedes of conoid and spheroid; for the diameter of the base one way, or the right angled conjugates of the base the other, with the height of either, will not limit a spheroid, as the diameter of the base and the height doth a sphere. This very rule Crowns all their endeavours; for before they had made a point bigger then any superfice, a line longer then infiniteness, a solid superfice, but now they are come to an unlimited solid.

6 In page 39. they write thus; But if such a so­lid have not its Zons made by circles or ellipses, but by four flat sides at right angles to the foresaid conju­gates, then it is a prismoid; nevertheless, the rules before prescribed, hold to all intents and purposes: that is false to all intents and purposes at the first appearance; for if two right lines be at right an­gles, [Page 10] and they be at right angles with four plains, those plains wil be the 4 sides of a Parallelepipedon, by the 2, 3, and 30 def. of 11 Euclid. a Parallele­pipedon being calculated gradually, can have but a first difference, and not a second and third: But this is like the rest of these Famous Geometers works. Our Master Geometer telleth his Reader thus, Most whereof have lain by me many years. And in the Title Page he saith, they are brief Collections from divers Authors: If so, why so many Fundamental Errors? Further, seeing M. Da­ry, and his Companions will assert any thing, and demonstrate nothing, except they are required in print; therefore I desire them to demonstrate these six following assertions of their own, and and shall call it