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

CHAP. V. Of the Conchoid, Cissoid, Cycloid, Quadratrix, &c.

Proposition XIX.

THE first conchoid of Nicomedes Bbb (Fig. 110) on both sides of the perpendicular cDb approaches nearer always to the directrix or horizontal line AE, and yet will never coincide with it, altho' it be conceived to be produced on both sides ad infinitum.

Demonstration.

For since only Db is perpendicular to AE, and all the rest ab are so much the more inclined to it by how much the more remote they are from the middle one Db, and all in the mean while are equal both to it and to one another, by Def. 13. it is evident that the points b and B will come so much the near∣er to AE, by how much the farther they recede from the mid∣dle line Db. And yet because the lines BAC and bac are all right ones, whose points A, a, are in a right line AE it is e∣qually as impossible that the point b or B, which is always in the conchoid should ever touch this right line, as it is impos∣sible that the point C should be in it, by vertue of the afore∣cited Def. Q. E. D.

Proposition XX.

YET no other right line can be drawn between the dire∣ctrix AE and the conchoid, but what will cut it if pro∣duced.

Demonstration.

For if such a right line be made parallel to AE, as GH, and you make, as DI to IC so Db to a fourth, which will be greater than IC, as Db is greater than DI, and consequently, if making that an interval you draw the circular arch from C, it will necessarily cut the line GH e. g. in G. Drawing there∣fore CaG. you'l have as DI to IC so aG to GC, i. e. to that fourth proportional before found, by vertue of Prop. 34. lib. 1. but as DI to IC, so was also Db to the same fourth by Con∣struc. Therefore aG and Db, which have both the same pro∣portion to the same quantity, are equal; and consequently the point G is in the conchoid by virtue of Def. 13. and conse∣quently the right line GH being produced will cut that pro∣duced also, on both sides, by the same reason. Much more will it cut it on either side if it be not parallel to the directrix AE, which is very obvious. Therefore no right line can be drawn between the conchoid, &c. Q. E. D.

CONSECTARY.

HEnce, besides orher Problems, that may be very easily solved, which requires, having any rectilinear angle gi∣ven ABC (Fig 143.) and a point without it, from that point to draw a right line DEF, so that part of it EF, which is intercepted between the legs of the angle, shall be equal to a given line Z. For if you draw the perpendicular DGH from the given point D through the nearest leg of the angle BC, and make GH equal to the given line Z, and from the center C at the interval GH describe the conchoid IHK, which will be necessarily cut by the other leg of the angle by vertue of the present Prop. e. g. in F, the line DF being drawn will

give the intercepted part = GH by the nature of the conchoid, and consequently = to the given line Z.

SCHOLIƲM.

BY means of this Consectary Nicomedes solves that noble Problem of finding two mean proportionals, after this way, which we will here shew from Eutocius, but drawn into a compendium, and somewhat changed as to the order. Let two given lines AB and BC (Fig. 144.) between which you are to find two mean proportionals, be joined together at right angles, and divide both into two parts in D and E, and ha∣ving compleated the rectangle ABCL, from L thro' D draw LG to BC prolonged; so that after this way GB may become = AL or BC: Having let fall a perpendicular from E cut off from C at the interval CF=AD the part EF, and having drawn FG make CH parallel to it; and lastly thro' the legs of the angle KCH draw the right line FHK, so that the part HK shall be equal to the line CF, by the preceding Consect. and also draw the right line KM from K thro' L to the con∣tinued line BA: All which being done, CK and AM will be two mean proportionals between AB and BC; which after our way we thus demonstrate: By reason of the similitude of the ▵ ▵ MAL and LCK

MA is to LC or AB as AL or BC to CK

b−eb−c−ec and moreover,

as MA to AD so GC to CK i. e. FH to HK

b−½eb−2c−ec by reason of GF and CH be∣ing parallel, by Consect. 4. Prop. 34. lib. 1. therefore since HK is = AD=½eb, FH will be = A=b, and consequent∣ly MD=FK, viz. both b+½eb, and the square of both =bb+eb+¼eebb=□EF+EK by vertue of the Pythag. Theor. Now if to these equal quantities you add the equal □□ DX and EC=¼cc, their sum, viz. □MD+□DX i. e. □ MX will be bb+ebb+¼eebb+¼cc, equal to the sum of these, viz. □EF+□EC i. e. □ CF (by the Pythag. Theor. or EX by Construct.) +□KX; whence these two things now follow: 1. That the lines MX and KX are equal. 2. If from those equal sums you take away the common quan∣tities

¼eebb+¼cc, the remainders will be equal, viz. bb+ebb =ecc+eecc; and (since the part taken away, viz. bb is ma∣nifestly to the other part taken away, viz. ecc as the remain∣der ebb to the remainder eecc, and the whole with the parts taken away and the remainders are in the same proportion by Prop. 26. lib. 1.) separately also bb will = ecc and ebb=eecc. But from the latter equation it follows that

as eb to ec so ec to b by vertue of the 19. Prop. lib. 1. AB to CK so CK to MA

and by the same reason it follows from the former Equation as ec to b so b to c

CK to MA so MA to BC i. e. CK and MA are two mean Proportionals between AB and BC. Q. E. D.

From which deduction you have also manifest the foun∣dation of that mechanical way, which Hiero Alexandrinus makes use of in Eutocius, lib. 2. of the Sphere and Cylinder, and which Swenterus has put into his practical Geometry lib. 1. Tract. 1. Prop. 23. when, viz having joined in the form of a rectangle the given right lines AB and BC (Fig. 145.) and continued them at the other ends, he so long moves the ruler in L, having a moveable center, backwards and forwards, 'till XK and XM by help of a pair of compasses are found equal. To which, another way of Philo's is not unlike, and flows from the same fountain, wherein, having made on AC a semicircle, the moveable ruler in L is so long moved back∣warks and forwards, until LM and NK are found equal: Which seems to Eutocius to be more accommodated to practice, and easier to be perform'd by help of a ruler divided into small equal particles.

Proposition XXI.

IF from any point of the other diameter in the generating cir∣cle e. g. from G (Fig. 111. n. 1.) you draw a perpendicu∣lar GE thro' the cissoid of Diocles. the lines CG, GE, GD, and GH will be continual proportionals.

Demonstration.

For since GE and IF, as right sines, and also GD and IC as versed sines of equal arches by the Hypoth. are equal; you'l have as ID to IF (i. e. CG to GE) so IF to IC (i. e. GE to GD) per n. 3. Schol. 2. Prop. 34. lib. 1. But GD is to GH as ID to IF (i. e. as GE to GD) by the forecited Prop. 34. lib. 1. Therefore CG to GE, GE to GD, and GD to GH, are all in the same continual proportion. Q. E. D.

CONSECTARY.

HEnce it was easie for Diocles to find two mean proportio∣nals x and y between two given right lines V and Z; (Fig. 146) for he made (having first described his curve DHB) as V to Z so CL to LK, and having drawn CKH to the curve, and thro' H the perpendicular GE, he had between CG and GH two mean proportionals GE and GD by vertue of the pre∣sent Prop. when in the mean while CG the first would be to GH the last, as CL to LK, i. e. as the first V to the last Z given by vertue of the Constr. Therefore nothing remain'd but to make, 1. as CG to GE so V to X; and lastly, as GE to GD so x to y.

SCHOLIƲM.

IT may not be amiss to mention here another way of find∣ing two mean proportionals between any two given lines by the help of two Parabola's, which Menechmus formerly made use of, viz. by joining at right angles the given lines AB and BC (Fig. 147.) and prolonging them as occasion shall require thro' E and D; and then describing a Parabola about BE as its axis, so made that BC shall be its Latus Rectum, and in like manner describing another Parabola about BD as its ax, that shall have AB for its Latus Rectum, and that shall cut the for∣mer in F: Which being done, the semiordinate FE (or BD which is equal to it) being drawn to the point of Intersection F, will be the two mean proportionals sought. For by ver∣tue of the fourth Consectary of Prop. 1. of this Book, DF or

BE is a mean proportional between AB and BD, and in like manner EF or BD is a mean proportional between BE and BC, and consequently as AB to BE so BE to BD, and as BE to BD so BD to BC; Q. E. D.

To this way of Menechmus that of Des Cartes is not unlike, which he gives us p. m. 91. except only that instead of two Parabola's, he makes use only of one and a circle in room of the other: In imitation of whom Renatus Franciscus Slusius has since shewn infinite methods of doing the same thing by help of a circle, and either infinite Ellipse's or Hyperbola's, in his ingenious Treatise which he thence names his Mesolabium.

Proposition XXII.

ANY semiordinate of the Cycloid as BF (Fig. 148.) or bf is equal to its corresponding Sine in the generating circle as BD, bd, together with the arch of that sine AD or Ad.

Demonstration.

For the motion of the point A describing the semi-cycloid AFE, by Def. 11. is compounded of the motion of the orb (or wheel) B along the semi-circle ADC, and of the motion of the centre along the right line BC equal to CE, and conse∣quently to the semi circle it self, or motion of the orb, There∣fore as the point A moving to E by the motion of the orb (or wheel) moved or was carried from the diameter AC thro' the whole semi circle ADC 'till it came to AC again, and by the motion of the Centre passes thro' the whole space BG or CE, which is equal to the semicircular arch; thus the same A, when come to F, will have describ'd the quadrant AD, by moving from the diameter AC the quantity of the sine BD, and moreover by the motion of its centre (which is equal to the motion of the Orb) moves from AC the space of DF: And so the semiordinate BF will be equal to the arch AD and to its sine BD taken together; and in like manner the semiordinate bf will be equal to the arch Ad and its sine bd, &c. Q E. D.

CONSECTARY I.

HEnce may be easily assign'd by help of the cycloid a right line equal to the semi-periphery or any given arch AD or Ad; viz. CE double, or taken twice for the whole circum∣ference, and single for the semi-periphery or half circumfe∣rence, DF for the quadrant Ad, df for the arch AD, &c.

CONSECTARY II.

WHerefore the quadrature of the circle may be geome∣trically obtain'd according to Consect. 2. of Def. 15. lib. 1.

CONSECTARY III.

IF you take Be, be, double of the sines BD, bd, &c. so that all the indivisibles bd taken together, may be to all the in∣divisibles be taken together as BD to Be, a curve described thro' the points e will be an ellipsis by Prop. 11. and the curviline∣ar space ADCeA will be equal to the semicircle ACDA.

CONSECTARY IV.

AND since DF (i. e. De+eF) is equal to the quadrant DA, by vertue of the present Prop. BD+FG will be also equal to the quadrant (because the whole BG or CE is = to a semicircle) and consequently eF and FG will be equal; in like manner since df both above and below is equal to the arch dA, below bd+fg will be = to the remaining Arch dC: and above bd+ef (i. e. df) will be equal to the equal arch dA. Therefore ef above and fg below are equal, and (since the same may be shewn of all the indivisibles of the same sort throughout) the trilinear figure FGE will be equal to the tri∣linear eFA.

Proposition XXIII.

THE cycloidal space is triple of the generating circle i. e. the semi-cycloidal space AECA is triple of the semi-circle ADCA.

Demonstration.

Since the parallelogram BCEG is equal to the whole circle by Consect. 2. of Def. 15. lib. 1. i. e. to the semi-ellipse AeCA, by the present construction the Trapezium CeGE will be equal to the quadrant of the ellipse or the semi-circle. But the trili∣near space FEG is = to the trilinear space FAe, by the fourth Consect. of the preced. therefore also the trilinear space AeCEA is equal to the semi-circle. Therefore the whole cycloidal space is equal to the three semi-circles. Q. E. D.

Or thus.

Since the whole parallelogram AE is equal to two circles and the semi ellipsis AeCA to one; the remaining space AeCEC to one circle, and its half AeGC to a semi-circle. But the tri∣linear space AeF is equal to the trilinear space FGE by Consect. 4. of the preced. Therefore the one being substituted in the other's place the trilinear space AFEC will be equal to the se∣mi-circle: Therefore the remainder of the Parallelogram, i. e. the cycloidal space AFECA will be equal to three semi-circles. Q. E. D.

SCHOLIƲM.

TO these short Demonstrations, which we confess we owe for the most part to Hon. Faber, we will subjoin another somewhat more prolix, but yet not unpleasant, which we find in Carolus Renaldinus, lib. 1. de Resol. & Compos. Math. p. 299. But here we will give it the Reader more plain, and free from all Scruples, and likewise much easier. It is perform'd in these inferences, 1. That the right lined Parallelogram AbaB (Fig. 149 n. 1) is equal to the curvilinear space AbdaBDA.

2. That as that is divided into two equal parts by its right li∣ned diagonal Aa, so likewise is this by the semi-cycloid Aaa, so that the right lined triangle AaB is equal to the curvilinear space AaaBDA. 3. Therefore the one as well as the other is equal to the generating circle; and consequently, 4 If to this curvilinear space there be added the semi-circle ADBA the semi-cycloidal space AaaBA will be equal to three semi-cir∣cles. The first is evident, while if you take from the right lined parallelogram the semi-circle ADBA on the one side, and on the other add the semi-circle abda, there will arise the curvi∣linear Parallelogram. The third is evident from Consect. 2. Def. 15. lib. 1. Because the line Ba is equal to the semi-peri∣phery, which multiplyed by the diameter BC gives the area of the circle. The fourth is self evident; and so there remains only the second to be demonstrated, viz. That the curvilinear paral∣lelogram is divided into two equal parts by the cycloid, i. e. that the external trilinear Figure AadbA is eqaal to the inter∣nal one AaaBDA; which may be thus shewn: Having divi∣ded the base Ba into three equal parts, and drawn thro' them three semi-circles, and moreover the transverse right lines Dd and Ee thro' the intersections of the sem-icircles and the cy∣cloid; it is certain from the genesis of the cycloid, by vertue of the Cons. of Def. 11. that as the right line a1 is a third part of the whole aB, so the arch 1a is a third part of the ge∣nerating periphery, and by the same reason the arch 2a two thirds, and so the remaining arch a11 also ⅓; insomuch that the first arch 1a, and the last a11, and consequently their right sines af, ag, and likewise their versed ones f1, g11, are equal, and so the curvilined partial Parallelograms, both above and below, all upon equal bases, and of the same height (viz. the two linear ones ae21 and da 11. 1.) are equal among themselves, and so likewise the two pricked or pointed ones Da 2B and aeb1. Wherefore if now the base Ba (n. 2) be con∣ceived to be divided into six equal parts, and having drawn semi-circles, and transverse lines thro' their intersections with the cycloid, the arches and 〈 math 〉〈 math 〉 will be 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉 of a semi-circle,

[illustration]

Pag. 227.

147

148

149

150

and so the versed sines of each, i. e. the altitudes of the corre∣sponding parallelograms will be equal, and consequently the parallelograms of the internal and external trilinear space that are alike noted or signed will be equal to each other. Now this inscription of curvilinear parallelograms always respective∣ly equal both in number and magnitude, since it may be con∣tinued in both the trilinear figures ad infinitum; it will evi∣dently follow, that the trilinear figures themselves, whose in∣finite inscripts are always equal, will be likewise equal to one another.

Proposition XXIV.

THE base of the quadratrix AE (Fig. 150) and the se∣midiameter of the generating quadrant AD and the qua∣drant it self BD are in continual Proportion.

Demonstration.

For the quadrant DB is to the radius DA as the arch IB to the perpendicular He by Consect. 1. Def. 16 and Ib is to He as Ab to Ae by Prop. 34. lib. 1. But the arch IB (if it be con∣ceiv'd to be less and less ad infinitum) will at length coincide with Ib, as ending in the same moment in the point B, where∣in He will end in the point E, and so Ae will end in AE and Ab in AB. Therefore at length DB will be to DA as IB (i. e. Ib) to He, i. e, as Ab to Ae, i. e. as AB (or DA) to AE. Q. E. D.

SCHOLIƲM I.

CLavius about the end of the sixth Book of Euclid, and o∣thers, demonstrate this indirectly by a deduction ad Ab∣surdum, or concluding the opposite much after this manner: If DA or AB is not to AE as DB to DA, suppose it to be so to the greater Af or the less Ae. In the first case therefore, because AB is to Af as DB to DA per Hypoth. i. e. as Kf to Af the quadrant Kf and the radius AB or DA will be equal. But as BD is to IB so is Kf to Hf by reason of the similitude of the arches; and as BD to IB so also DA (or =Kf) to the sine He, by

Consect. 1. Def 16. Therefore the sine He and the arch Hf (to which the same Kf bears the same proportion) will be e∣qual; which is absurd. In the latter case, because AB would be to Ae as DB to DA by the hypoth. i. e. as Le to Ae, the quadrant Le and the radius AB or DA would be again equal. But as BD is to IB so is Le to Me by reason of the similitude of the arches; and as BD to IB so also is DA (ie=Le) to He by Consect. 1. Def. 16. Therefore the tangent He and the arch Me (to which the same Le bears the same proporti∣on) will be equal, which is again absurd. Wherefore BD is to DA, not as DA to a greater Af or a less Ae; therefore as DA to AE. Q. E. D.

CONSECTARY I.

WHerefore it is evident from what we have deduced, if by means of the base of the quadratrix AE you draw a quadrant, the side of the quadratrix DA will be equal to it, and consequently double of the semi periphery, and quadruple of the whole periphery.

CONSECTARY II.

IT is evident also that you may obtain a right line equal to the quadrant DB of any given circle, if, having described a quadratrix, you make as AE to AD so AD to a third equal to the quadrant DB: Which third proportional taken four times will be equal to the whole periphery.

CONSECTARY III.

YOU may also obtain a right line equal to any less arch, if you make, as DA to He so a third proportional found (i. e. the quadrant DB) to a fourth, by vertue of Cons. 1. Def. 16.

CONSECTARY IV.

THE quadrature of the circle therefore, by vertue of Cons. 2. Def. 15. lib. 1. as likewise the trisection of an angle, by vertue of Consect. 2. Def. 16. lib 2. may be Geometrical∣ly obtain'd, if the Quadratrix might be number'd among Geo∣metrical Curves.

SHCOLIƲM II.

CLavius was also of this Opinion in the book afore menti∣oned, who thought that if the quadratrix be excluded out of the number of geometrical curves, by the same reason you may also exclude the ellipse, parabola, and hyperbola, since they as well as this are commonly described thro' innu∣merable points. But by that great Man's leave, we may de∣ny this consequence, by the same reason as Des Cartes has de∣ny'd the converse of it in his Geom. p 18. and 19. by vertue of which he suspects the ancients took the conick sections, &c. for mechanick or non-geometrick lines, because they did the spiral, quadratrix, &c. for such. But this is the difference between the description of the quadratrix and the conick secti∣ons thro' points, that all and every of the points of the conick sections, relating to any given point of the axis, may be geo∣metrically determin'd; but all the points of the quadra∣trix promiscuously related to any point of the generating qua∣drant, cannot be geometrically determin'd, but only those which respect some certain point, from which the quadrant may be divided into two arches of known proportion. For if, e g. in the quadrant BD the point X be given at pleasure, it will be impossible by Clavius's Rule to define a point of the quadra∣trix answering to it, because the proportion of the arches DX and BX is unknown, and consequently neither can a propor∣tional section of the right line AD be made: Not to mention that the last point E (which is the primary and most necessary one to the quadrature) even by Clavius's own confession cannot be geometrically defined. We may pass the like judgment on Archimedes's spiral and such like curves, which are conceiv'd to be described by two motions independent on one another; as

will be manifest to any one who compares the genesis of the spiral with that of the quadratrix and what we have hitherto said. Whence neither will Monantholius's trisection of a gi∣ven angle (which he essays) by means of a spiral be enough geometrical; which in his Book de Puncto, Cap. 7. p. 24. he attempts to perform thus: To the centre of a described spi∣ral and its first helical or spiral line BA (Fig. 151.) he applies the angle ABC equal to the given one abc; then having drawn circles thro' F and A where the legs of the angle cut the spi∣ral, he divides the intermediate space DA into three equal parts in 1, and K: And then thro' these points he draws circles cut∣ing the helix in L and M; and lastly having drawn BLN, BMO, he easily demonstrates from the genesis of the spiral that the arches AO, ON, NC are equal. And so after the same manner not only any angle or arch, but the whole peri∣phery may be geometrically divided into as many parts as you please; only supposing that this spiral line may be numbred among geometrical ones; as we have heretofore hinted that the cycloid, conchoid, cissoid, and logarithmical curve, &c. might be; and we have above sixteen Years ago declared our opinion for it in our German Edition of Archimedes; and now are therein confirm d by those celebrated Mathematicians Leib∣nitz, Craige, &c. who number lines of this kind, altho' they cannot be expressed by our common equations, among geome∣trical ones, notwithstanding the contrary opinion of Des Car∣tes, &c. because they admit of equations of an indefinite or transcendent degree, and are capable of a Calculus as well as others, tho' it be of a nature and kind different from that commonly used. See the Acta Erud. Lips. ann. 84. p. 234. and ann. 86. p. 292. and 294.