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

CHAP. III. Of Conoids and Spheroids.

Proposition XIV.

A Parablick Conoid(α) 1.1 is subduple of a Cylinder, and in sesquialteran reason (or as 1 ½) of a cone of the same base and altitude.

Demonstration.

Because in the parabola the □ AD (Fig. 137.) is to the □ SH, as BD to BH, i. e. as 3 to 1, and so to the □ TI as BD to BI, i. e. as 3 to 2, by Prop. 4. of this; it is evident that these squares of SH and TI and AD and consequently of the whole lines also Sh, Ti, AC, and the circles answering to them will be in arithmetical progression, 1, 2, 3; and more∣over if there are new Bisections in infinitum, as the abscissa's so also the squares and circles of the ordinates, by vertue of the aforesaid fourth Prop. will always be in arithmetical Progres∣sion 1, 2, 3, 4, 5, 6, &c. It is evident that an infinite series of circles in the conoid, consider'd as its indivisibles, will be to a series of as many circles equal to the greatest AC, i. e. the co∣noid to the cylinder AF as 1 to 2, or as 1 ½ to 3, by Consect. 9. Prop. 21. or Consect. 4. Prop. 16. lib. 1. but to the same cylinder AF the inscribed cone ABC is as 1 to 3, by Prop. 38. lib. 1. therefore the cylinder, conoid and cone are as 3, 1 〈 math 〉〈 math 〉 and 1. Q.E.D.

Proposition XV.

THE half of(α) 1.2 any Spheroid, or any o∣ther segment of it is in subsesquialteran pro∣portion to the cylinder, and double of the cone having the same base and altitude.

Demonstration.

Having divided the altitude BD (Fig. 138.) e. g. into three equal parts, because in the ellipse as well as in the circle the square of AD is to the square of SH as the □ GDB to the □ GHB, i. e. as 9 to 5, and so to the square TI as 9 to 8, by Consect. 1. Prop. 5. of this; and in like manner if you make new bisections, the squares (and consequently the circles) of the ordinates go on or decrease by a progression of odd num∣bers, as 36, 35, 32, 27, 20, 11, and so ad infinitum, the bisections being continued on; as we have shewn in the sphere and circumscribed cylinder Prop. 39. lib. 1. and it will neces∣sarily follow here also (by vertue of Consect. 12. Prop. 21.) that the whole cylinder will be to the inscribed segment of the spheroid, as 3 to 2; and since the same cylinder is to the cone ABC as 3 to 1, also the segment of the spheroid will be to the cone as 2 to 1. Q. E. D.

Proposition XVI.

AN hyperbolical Conoid(α) 1.3 is to a cone of the same base and altitude, as the aggre∣gate of the ax of the hyperbola that forms it and half the Latus Transversum, to the aggregate of the said axis and Latus Transversum.

Demonstration, containing also the Invention of this Proportion.

Make (in Fig. 139.) CE=a, EF=b, OE=c; then will CF=a+b. Since therefore as CE to OE so CF to FQ 〈 math 〉〈 math 〉 the □ EO will = cc and □ FQ= 〈 math 〉〈 math 〉.

But as these squares so also are the circles of the lines EO and FQ to one another, by Prop. 32. lib. 1. and so the cone COP

will be as 〈 math 〉〈 math 〉, and the cone CQR as 〈 math 〉〈 math 〉 (viz. by multiplying the third part of the altitude CF by the base FQ:) Having therefore substracted the cone COP from the cone CQR, there will remain the truncated cone QOPR 〈 math 〉〈 math 〉, and from this solid truncated cone having further substracted the hollow truncated cone, which the space EHRP produced in the genesis of the conoid (and which ac∣cording to Consect. 2. Definit. 9. is as bcc) there will remain the hyperbolical conoid 〈 math 〉〈 math 〉 i. e. (by substituting now the values of the ax or abscissa EF, and of half the Lat. Transv. EC, and of the conjugate diam. OP, &c. found in the de∣monstrations of the preceding Chapter, viz. 〈 math 〉〈 math 〉 for a, eb for b, and √oocd for c or oocd for cc) the hyperbolical conoid will come out 〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉. But the cone GEH (multiplying the third part of EF into the □ GH, i. e. ⅓eb into 4oecd+4eecd) is as 〈 math 〉〈 math 〉. Therefore the conoid is to the cone as 6eebocd+4e{powerof3}bcd to 4eebocd+4e{powerof3}bcd, i. e. (dividing on both sides by 4eecd) as ½ob+eb to ob+eb. Which was to be found and demon∣strated.

SCHOLIƲM.

IF any one had rather proceed herein by indivisibles, as in the precedent Prop. having divided the ax EF (Fig. 140.) again into three equal parts, and assuming the values of the lines determined in the hyperbola, viz. eb for the abscissa EF, ob for the transverse ax, 〈 math 〉〈 math 〉 for the Latus Rectum, oecd+eecd for the square of the semiordinate FG, &c. the lowest and greatest circle of the diameter HG will be as oecd+eecd, and, if you make

as the Latus Transv. to the Latus Rectum, so the □ DfE ob−〈 math 〉〈 math 〉 made of ob+⅔eb into ⅔ eb (i. e. ⅔oebb+〈 math 〉〈 math 〉eebb) to a fourth; there will come out ⅔oecd+〈 math 〉〈 math 〉eecd for the second circle of the diam. hg; and by the same inference (as ob to 〈 math 〉〈 math 〉 so ob+⅓eb into ⅓ eb to a fourth) for the third circle of the diameter HG ⅓oecd+〈 math 〉〈 math 〉eecd; so that these indivisibles [for which here and in the precedent also the partial circumscribed cylinders may be assumed] proceed in a double series of numbers, the first in a simple arithmetical progression 3, 2, 1, the latter in a du∣plicate Arithmetical progression of squares 9, 4, 1; and the same if you make further new bisections, will necessarily hap∣pen ad Infinitum, (the former numbers e. g. in the first bise∣ction will be 〈 math 〉〈 math 〉 ⅚ 〈 math 〉〈 math 〉 oecd the latter 〈 math 〉〈 math 〉 eecd, &c) it is manifest from the consectaries of Prop. 21. lib. 1. that the whole cylinder HK will in like manner be expressed by a double series of parts answering, in numbers to the indivisibles of the conoid made by any bisection, but in magnitude to the greatest of them all, and in the sum of its first series of parts will be to the sum of the first in the conoid, both being infinite, as 2 to 1 or 3 to 1 ½ oecd, by Consect. 9. of the said Prop. 21. and the sum of its latter to the sum of the former in the conoid will be as 3 to 1 eecd and so the whole cylinder to the whole conoid as 3 oecd+3 eecd to 1 ½ oecd + eecd i. e. (dividing by ecd) as 3o+3e to 1 ½o+e i. e. mul∣tiplying both sides by b) as 3ob+3eb to 1 ½ob+eb; and consequently the cone (which is ⅓ of the cylinder) to the co∣noid as ob+eb to 1 ½ob+eb. Q.E.D.

CONSECTARY.

HEnce also appears the proportion of the hyperbolick co∣noid to a cylinder of the same base and altitude, which we did not express in the Prop. viz. as the aggregate of the ax and half the Latus Transversum to triple the aggregate of the said ax and Latus Transversum.

Notes

• (α) 1.1

  Archimed. Prop 23. and 24. (al. 26. and 27.)
• (α) 1.2

  Archim. 29. and 30. (al. 32. and 33.)
• (α) 1.3

  Archi••. Prop. 27. and 28. (al. 30. and 31.)