Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

THE SECOND BOOK. (Book 2)

SECTION I. Containing DEFINITIONS.

Definition I.

IF a Cone ABC (Fig. 101.) be conceived to be cut by a plane at right angles to the side of the cone, e. g. BA; the Plane EFGHE arising by this section, and terminated on the external surface of the cone by the curve line HEG, &c. was anciently by Euclid, Archimedes, &c. called the Conick Section; and if the sides of the cone AB and BC made a right angle at B, as n. 1. the section was particularly called the Se∣ction of a right-angled Cone; but if it made an obtuse angle, as n. 2. it was called the Section of an obtuse-angled Cone; if, lastly, it made an acute one, as num. 3. it was called (3) the Section of an acute-angled Cone.

Definition II.

BUT afterwards their Successors, particularly Apollonius Pergaeus, found from the properties of these Curves, which their Predecessors had happily discovered, that the same (all of them) might be generated in one and the same cone whether right-angled, obtuse-angled, or acute-angled, if the section EF (Fig. 102.) is made in the first case parallel to

the opposite side BC; in the second case, if it meet that side produced upwards; for the third, when it meets downwards with the diameter of the base AC produced to D. And to give new names (for the old ones would do no more now) to these Sections, to distinguish them one from another, nomi∣nating them from their Properties hereafter demonstrated, they called the first a Parabola, the second an Hyperbola, the third an Ellipsis.

Definition III.

BUT it is evident, that only the plane making the secti∣on of the second case, being according to the line FED produced or carried on, (Fig. 103.) will fall upon the verti∣cal Cone aBc comprehended under the sides AB, CB, &c. continued onwards, and there produce another Section oppo∣site to the former; whence these, viz. GEHG and gehg are called opposite Sections.

Definition IV.

BEsides these names of the sections, there are several others made use of to denote various lines drawn and considered both within and without those sections, the chief whereof we shall here explain. And first of all, in general the line EF so let fall thro' the middle of the section from its top E (which is called the Vertex of the section) to the diameter of the base of the Cone AC (produced if occasion be) that it shall bisect the line GH and all others parallel to it, is called the Di∣ameter of that Section; and particularly it is called the Axis of the Section if it cuts them at right angles or perpendicularly; as also they name those lines GH, KN, &c. which are cut in∣differently by the diameters, but at right angles by the Axis, those, I say, they call Ordinates, or Ordinate Applicates, and their halves, FG, IK, &c. Semiordinates, (or some also call the latter Ordinates, and the former double Ordinates) and the parts of the Ax or Diameter EF, EI, &c. are called Abscis∣sa's (by some intercepted axes and diameters)

Definition V.

PArticularly in the hyperbola they call the continuation of the ax or diameter ED 'till it meet the opposite side cB, i. e. to the vertex of the opposite section, the Latus transver∣sum of the Hyperbola, to which there answers in the Ellipsis the axis or longest diameter, and so by latter Authors is called by the same name, but by Apollonius the transverse Ax or the transverse Diameter, as also the shortest ax or diameter is cal∣led the Conjugate, and its middle point O is called the Center of the Section or of the opposite Sections.

Definition VI.

THey called also a certain line EL (Fig. 101.) by the name of Latus Rectum, which is particularly to be found in all the sections, as we shall hereafter shew: And be∣cause this Latus Rectum is a sort of a Rule or Measure, ac∣cording to which the squares or powers of the ordinates used to be estimated or valued (as we will shew in its proper place) therefore the Ancients used to call it by a Periphrasis Linea se∣cundum quam possunt Ordinatim applicatae, or the measure of the powers of the Ordinates; by some latter Authors it is cal∣led the Parameter. Now a mean proportional PQ found be∣tween this Latus Rectum and the Latus Transversum (Fig. 104 n. 1. and 2) (see also hereafter Consect. 2. Prop. 8.) and drawn thro' the centre O parallel to the Ordinates is cal∣led the second Axis or Diameter, or the Conjugate of the Hy∣perbola.

Definition. VII.

NOW if we conceive the diameter or conjugate ax PQ brought down to the Hyperbola so that its middle point O shall touch its vertex in E, and from the center O you draw the right lines OR, OS, thro' the ends of this tangent line p and q, these are the lines which Apollonius, Prop. 1. lib. 2. de∣monstrates, that tho' by being continued, they always ap∣proach nearer and nearer to the curve GEH, and come so

[illustration]

Pag. 156.

104

105

106

107

much the nearer by how much the farther they are continued, yet they will never concur with it or touch it, for which rea∣son they are called Asymptotes or non-coincident lines; and by some the Intactae. Which non-coincidency appears most manifestly where the hyperbolical section of the cone is made parallel to the triangular section thro' the axis of the cone ABC (n. 3.) along the line e. f. parallel to the ax BF. For if we conceive the hyperbola geh to move forwards always pa∣rallel to it self, according to the direction of the equal and pa∣rallel lines gG, fF, and hH, 'till it stands in the position GEH; it is manifest that the curve line GEH is distant on both sides from the right lines BC, BA, the length of the versed sine of the equal arches hC and gA in the circumference of the circu∣lar base, while in the mean time it is evident that they ap∣proach nearer and nearer to them. So that hence there flow the following

CONSECTARYS.

I. IN this case the sides of the cone are the Asymptotes of the hyperbola, while it is manifest, that the point B is its centre, and EB half the transverse diameter; which appears from n 1. and 2. of the pres. Fig. for the section ef being made parallel to the the ax of the cone DF by def. 5. de (which in the case n. 3. would coincide with dq) is the trans∣verse diameter, but the triangles dpq and OpE are equiangular, and consequently as pE is to ½pq, so is OE to ½dq.

II. The lines AG and HG (num. 3.) are equal, as being the versed sines of equal arches; and in like manner (n. 1. and 2.) RG and HS are equal, since FR and FS as well as Ep and Eq are so also (for

• as OE to Ep, −Eq
• so OF to FR −FS

and the semiordinates FG and FH are also equal.

III. Consequently □ □ of RG into GS and of HS into HR are equal, &c. all which hereafter we will more univer∣sally demonstrate.

Definition VIII.

IF a Parabolick plane(α) 1.1 HEGFH (Fig. 105. n. 1.) together with a triangle HEG inscribed in it, and a rectangle HL circumscri∣bed about it, be conceived to be moved round about the ax EF on the point F; it will be evident that by the triangle there will be generated a cone, by the rectangle a cylinder, and by the parabola a parabolick solid, which with the comprehended cone, and the comprehending cylinder, will have the common base HIGK and the same altitude EF, and was by Archimedes named a Parabolick Conoid.

Definition IX.

IF moreover an(α) 1.2 hyperbolick plane HEGFH (n. 2.) with the inscribed trian∣gle HEG, and another circumscribed one ROS made by the Asymptotes OR, OS, be concei∣ved to be turned about the common ax OEF on the point F; it will be evident that there will be described by the inscribed triangle a cone comprehended within side, and by the hyper∣bola an Hyperbolical Conoid upon the same base HIGK and of the same heighth EF; and by the ▵ ROS another cone which Archimedes calls the comprehending cone, whose base is RTSV, and its altitude composed of the axis of the hyperbo∣la EF and half the transverse ax OE (which Archimedes cal∣led the additament of the ax of the hyperbola) and which we may commodiously divide into two parts, viz into the cone OPMQL, whose base has for its diameter the conjugate ax PQ, and its altitude equal to half the transverse ax; and into a Curti-cone or detruncated cone terminated by the two bases PMQL and RTSV, but answering in altitude to the conoid and inscribed cone: From which, as comprehending it, if you take away the included conoid, there will remain the hollow curticone terminated below by the Annulus or ring RGIVHSKT and above by the circular base PMQL, and generated in the circumvolution by the intermediate lines EP, GR, &c. or the mixtilinear plane EGRP. Now if we suppose instead of

this comprehending cone a circumscribed cylinder on the same base and of the same altitude with the conoid and included cone, you'l have every thing like as in Def. 8.

CONSECTARYS.

NOW if we suppose the 1. case of Def. 9. to be expres∣sed by the Fig. of Def. 7. n. 3. and conceive the pre∣sent figure brought thence to be turned round about the ax BEF (Fig. 106.) we may deduce these following things in the room of Consectarys for the foundation of our future de∣monstrations.

I. The lines EQ, RS, HC, &c. of the mixtilinear space comprehended between the hyperbola and the Asymptotes (viz. the excess of the ordinates) altho' they are unequal, and by descending always grow less, yet in this circumvolution they will describe equal circular spaces, viz. EQ a whole circle, (or circular plane) but RS and HC, &c. circular Annuli or rings all of the same bigness; which will thus appear to any one who compares this figure with the former: Since the spa∣ces generated by the lines EQ, FC, &c. are as the squares of those lines, and the □ of Fh or FC exceeds the square of fh or FH by the excess of the square Ff or Ee or EQ, conse∣quently the quantity generated by FC will exceed that gene∣rated by FH the excess of that generated by EQ; and also that generated by FH by the excess of that generated by HC; it is manifest that the circle generated by EQ will be equal to the annular space generated by HC; and the same will in like manner be evident of any spaces produced by RS.

II. Therefore the hollow detruncated cone generated by the space EHCQ according to Def. 9. will be equal to a cylinder generated by the rectangle FIQE; for all the indivisibles of the one, are equal to all the indivisibles of the other by Con∣sect. 1.

Definition X.

IF, lastly,(α) 1.3 an elliptical plane be turned about either of the axes, viz. either the lon∣gest DE (Fig. 107. n. 1) or the shorter AB (n. 2.) there will be thence formed an elliptical solid, called by Archimedes a spheroid; which in the first case will be an oblong or erect one, in the other a flat or depres∣sed one: And it is self evident, that if before this circumvolu∣tion of the ellipsis, there be inscribed in one of its halves a triangle, and also a rectangle circumscribed about it, having the same altitudes and bases with the semi ellipse, there will af∣terwards in the circumvolution be described by the triangle a cone, by the rectangle a cylinder, to which afterwards we will also compare the half spheroid; as also both the conoids with their inscribed and circumscribed cones and cylinders.

Definition XI.

IF upon the right line AE (Fig. 18.) you conceive a wheel or circle to rowl, until its point A, with which it touches the said line, come to touch it again in E; the circle wil mea∣sure out the line AE equal to its periphery; but the point A by its motion will describe the curve line AFE, which is cal∣led a Trochoid or Cycloid; and the area which this curve with the subtense AE comprehends, is named the Cycloidal Space; and the circle by whose motion they are determined is called the generating Circle.

CONSECTARY.

IT is evident from the genesis of this curve that the descri∣bing point a will always be as much distant in the circle from the point of contact d or c, as the point A in the right line passed over AE, is from the same point of contact, i. e. if the point d is distant from A the fourth part of the line AE, the arch da will also be the fourth part of the circle considered in this second station; and the point c being distant from A half of the interval AE, the arch ca will be also half of the

[illustration]

Pag. 160.

108

109

110

111

circle, and so the point a coincide in the curve with F: And when the point e is distant from E only an eighth part of the whole line AE, the arch ea will also be the eighth part of the whole circle.

Definition XII.

IF the right line BA (num. 1. 109.) one end at B remain∣ing fixed, be moved round at the other end with an equal motion from A thro' C, D, E to A back again, and in the mean while, there be conceived another moveable point in it to move with an equal motion along the line BA from B to A, so that in the same moment wherein the moveable point A absolves one revolution, the other moveable point shall also have passed thro' its right lined way, coinciding with the point A retur∣ned to its first situation; this extremity A by its revolution will describe the circle ACDEA, and that other moveable point another curve B, 1, 2, 3, 4, &c. which with Archi∣medes we will call a Helix or spiral Line, and the plane space comprehended under this spiral line and the right line BA in the first station is called a spiral space. Now if we suppose, e. g. the right lined motion of the point moving along BA to be twice slower than in the former case, so that (see num. 2.) in the same time that the point A makes one whole Re∣volution, the other moveable point shall come to F, making half the way BA, and then at length shall concur or meet with the extremity, when that shall have performed the other revolution; and so there will be described a double spiral line, whose parts with Archimedes we will so distinguish, that as he calls the part of the right line BF, passed over in the first revolution, simply the first line, and the circle made by the right line BF the first Circle; so we will call that part of the curve which is described in that time or revolution B 136912 the first Helix or the first Spiral, and the area comprehended un∣der it the first spiral space: And, as the other part of the right line FA passed over in the other revolution is called the second line, and the circle marked out by the whole line BA the se∣cond Circle; so the curve described in the mean while 12, 15, 18, 24, may be called the second spiral line, and the space

comprehended under it the second spiral space, and so onwards From these Definitions there flow the following

CONSECTARYS.

I. THE lines B 12, B 11, B 10, &c. drawn out, and making equal angles to the first or second spiral (and after the same manner(α) 1.4 B 12, B 10, B 8, &c. or B 12, B 9, B 6, &c.) are arithmeti∣cally proportional, as is evident.

II. The lines drawn out to the first spiral as B 7, B 10, &c. are one among another as the arches of the circles inter∣cepted between BA and the said lines(β) 1.5 B 7, B 10, &c. which is also evident to any one who considers what we did suppose; for in the same time as the end A passes over seven parts of the circle, the other moveable point will also run over seven parts of the right line BA, &c.

III. Lastly, The right lines drawn from the initial point(γ) 1.6 B to the second spiral e. g. B 19 and B 22 (num. 2.) will be one to ano∣ther as the aforesaid arches together with the whole periphery added to both sides: for at the same time the extremity A runs thro' the whole circle or 12 parts and moreover 7 parts (i. e. in all 19 parts) in the same time the other moveable point passes through 12 parts of the right line BA (in this case di∣vided into 24 parts) and moreover 7 parts, that is, in all 19; and so in the others.

Definition XIII.

IF a right line BAF be conceived so to move within the right angles ADC, CDE, that on the one hand a certain point C fixed in one leg of the Norma or ruler may always glide along, and on the other hand a certain moveable point A may always run along the other side of the Norma (which complicated motion of the describing rule BAF, after what way it may be organically procured, may be seen by

[illustration]

Pag. 163.

112

113

114

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the 110th. Figure;) by the extreme point of the moveable line B there will be described a curve called by its Inventor Nicomedes the first Conchoid, whereof this is a property, that the right lines CB, Cb, drawn from its centre C to its Ambi∣tus or curvity are not themselves, as in the circle, equal, but yet have all the parts, AB, ab, intercepted between the curve, and the directrix AE are equal; as is evident from its genesis.

Definition XIV.

IF, the diameters of a circle being AB and CD (Fig. 111. n. 1.) cutting one another at right angles, you take BE or Be and BF or Bf equal arches, and from E or e you draw the perpendiculars EG or Eg, and through these from D to F or f you draw the transverse lines DF or Df; the several points of intersection H, h, &c. decently connected together will exhi∣bit the curve line Dh, HhB (which may be continued also without the circle if you please, and) which is commonly at∣tributed to Diocles, and called a Cissoid.

Definition XV.

IF, having divided the right line AE (Fig. 112.) into the equal parts AB, BC, &c. from the points of the division A, B, C, &c. you draw the parallel lines Aa, Bb, Cc, &c. in geometrical progression (as e. g. let Aa be 1, Bb 4, Cc 16, Dd, 64, &c. or Bb 10, Cc 100, Dd 1000, Ee 10000, &c.) and further bisecting AB, BC, &c, F, G, H, I, you let fall mean proportionals between the next Collaterals Ff, Gg, Hh, Ii, &c. and continue to do so till the parallels are brought near to one another; the curve line drawn thro' the extremities of these lines af, bg, ch, die, will be the logarithmical line of the moderns, whose properties and uses are very excellent.

SCHOLIƲM.

AMong those uses, that is none of the least, from which this curve borrows its name, viz. in shewing the na∣ture and invention of Logarithms. For, e g. 1. If this line was accurately delineated in a large space, the parts AB,

BC, &c. being taken so big, that they might be subdivided not only into 100 or 1000 but into 10000 or 100000 parts; making AB 100000 (and so A 00000) AC would be 200000, AD 300000, &c. while in the mean while there answer to these as primary Logarithms in arithmetical pro∣gression the geometrical proportional numbers, Aa 1, Bb 10, Cc 100, Dd 1000, Ee 10000, &c. Whence, 2. Its Loga∣rithm may be assign'd to any given intermediate number, e. g. to the number 982, for having cut off this number from Dd by a geometrical scale on the line DM, if you draw Mn pa∣rallel to AD, and nN parallel to DM, it will give AN on the same scale, viz. the Logarithm sought, and reciprocally. But if, 3. it seem difficult to delineate a Figure so large, yet at least the clear conception of such a delineation evidently shews the arithmetical method, which those ingenious Men have made use of, who have made the tables of Logarithms with a great expence of Labour and pains, viz. by finding continual mean proportionals, arithmetical ones between any two Logarithms already known, and geometrical ones between two vulgar numbers answering to them, &c. by comparing what we have noted in Schol. 2. Prop. 20. Lib. 1. And we will note, 4 out of Pardies, that, since the Logarithms of numbers di∣stant from one another by a decuple proportion, differ by the number 100000, having found the Logarithms of all the numbers from 1000 to 10000 you will at the same time have all the Logarithms of all the other numbers that are between 100 and 1000, between 10 and 100, and between 1 and 10, only changing the characteri∣stick, and lessening it in the first case by unity, in the second by 2, in the third by 3; as e. g. if of the number 9, 900 you had found the Logarithm 399, 563, the Logarithm of the subdecuple number 990 would be (viz. substracting from the former 100000) 299, 563. and the Logarithm again of this subdecuple of this 99 would be 199, 563, &c. Thus in the Chiliads of Briggs to the number

99000	Answ.	4,99563,51946
9900	—	3,99563,51946
990	—	2,99563,51946
99 lastly	—	1,99563,51946.

But there will not arise such advantage for making Loga∣rithms by this observation as it may at first sight seem to pro∣mise, because there are 9000 numbers between 1000 and 10000 whose Logarithms must be found also, and but 900 between 100 and 1000, and but 90 between 10 and 100, and but 9 between 1 and 10, and so in all 999, which is not the ninth part of the former.

Definition XVI.

IF the radius AD (Fig. 113.) be conceived to move equal∣ly about the point A through the periphery of the quadrant DB, while in the mean time the side of the square DC re∣maining always parallel to it self, descends also with an equal motion thro' DA, so that in the same moment the radius AD and the aforesaid side DC shall fall upon the base AB; or (if any one should think that this way the proportion of a right line to a circular one is supposed by a sort of Petitio Principii or begging the question) the right line DA as well as the quadrant DB being divided into as many equal parts as you please (e. g. here both of them into 8) and drawing thro' these from the center A so many Radii and thro' them paral∣lel lines; the points of intersection being orderly connected to∣gether will exhibit a curve line, whose invention is attributed to Dinostratus and Nicomedes in the fourth Book of Pappus A∣lexandrinus, and which from its use is called a Quadratrix, it having among the rest this property, that from AB it cuts off a part AE, which is a third proportional to the quadrant DB and its radius DA; which hereafter we will demonstrate. In the mean while from this description of it, you have these

CONSECTARYS.

I. IF thro' any point H assumed in the Quadratrix you draw the radius AHI, and from the same point the perpendi∣culars Hh and He, the whole arch DB will always be to the part IB cut off, as the whole line DA to the part hA cut off, or He equal to it.

II. Consequently therefore any given arch or angle of the quadrant e. g. IB or IAB may by help of the quadratrix be divided into three equal parts or as many as you please, or in what proportion soever you will; while having drawn the radius AI, the perpendicular Ha let fall from the point of the quadratrix H, may be divided into three or as many equal parts as you please, or in any proportion whatsoever, and thro' these sections radius's drawn to divide the arch.

Notes

• (α) 1.1

  Archim. de Conoid. & Spher. Def. 1.
• (α) 1.2

  Archim. de Conoid. & Spher. Def. 3.
• (α) 1.3

  Archim. de Conoid. & Spher. Def. 6.
• (α) 1.4

  Archim. Prop. 2. of Spirals.
• (β) 1.5

  Archim. Prop. 4.
• (γ) 1.6

  Archim. Prop. 15.