The Geometriae Pars Universalis (GPU) by the Scottish mathematician James Gregory is a 17th century mathematics text which uses geometrical techniques to solve a variety of calculus problems, such as finding tangents, areas, and volumes of revolution. Buried without fanfare right in the middle of the GPU (the 35th of 70 propositions in the work) is a seemingly innocuous geometry result about solids of revolution:

Each solid of revolution is equal to a right cylindrical figure whose base is the figure out of the rotation of which the solid is produced and whose altitude is the circumference of a circle in which the center of gravity of the figure is revolved.

For our generation of mathematicians, solids of revolution serve primarily as a testing ground for the techniques of integral calculus. So on first glance we might miss the significance of a result which simply gives an equality between the volumes of two three-dimensional objects. However, since the volume of a right cylindrical figure is the area of its base times its height, another way of interpreting Gregory's result is

This rephrasing of Gregory's Proposition 35 may be familiar to those who have seen second semester calculus. It states that the volume of each solid of revolution is equal to the area of its base multiplied by the circumference of the circle in which the center of gravity of that figure is revolved. This is the Theorem of Pappus (or the Pappus-Guldin Theorem). Gregory's geometrical approach toward proving this result and just why this result ended up in Gregory's text in the first place are the subjects of this article.

The Pappus-Guldin Theorem is simultaneously one of the last great results in Greek mathematics and one of the first novel results in the 16th and 17th century renaissance in European mathematics. It was first stated by Pappus of Alexandria, who flourished around 300-350 CE, in Book Seven of his Collection. In the midst of a discussion of the Conics of Apollonius, Pappus launches into an emotional lament on the state of mathematical learning in his time:

When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers . . . [1, p. 122]

He then presents the following as evidence that the glory of Greek mathematics has not faded completely during his time:

The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them . . .

Several additional results follow, but all are variations on the first one.

Like Gregory's statement of his result, it's necessary to peel away some geometrical ideas (this time in Euclidean proportion theory) to find the familiar result. If AB is a figure with center of gravity a which is revolved around the axis CD, let radius(a) denote the distance between a and CD. Similarly, if EF is another figure with center of gravity e which is rotated around the axis GH, let radius(e) denote the corresponding distance.

Let area(AB) and area(EF) denote the areas of the two planar figures AB and EF, and let rev(AB) and rev(EF) denote the volumes of the solids obtained by revolving AB and EF around the axes CD and GH, respectively. If, as is customary, we interpret ratios as fractions and compounding of ratios as multiplying fractions, then Pappus' result becomes:

Multiply the top and the bottom of the equation by 2π to get

Here circum(a) and circum(e) denote circumferences of a circle of radius(a) and radius(e), respectively. This is not quite our formula, but to a mathematician brought up in Euclidean proportion theory this proportional expression of the result is perfectly natural and analogous to the formulas for areas and volumes occurring in Euclid's Elements (e.g., Euclid VI.23, or Euclid XII,18).

The contribution of Paul Guldin (1577-1643) to the Pappus-Guldin Theorem occurs toward the end of a long road of re-discovery and invention related to centers of gravity. Archimedes had initiated the classical study of centers of gravity in the two books On the Equilibrium of Planes [2]. Given its rather marginalized status in today's mathematics curriculum, it might be surprising to learn that finding centers of gravity was an important research topic in mathematics in the 16th and 17th centuries which was completely on par with computation of areas, volumes, and tangents. The translation of Archimedes' work on centers of gravity by Frederico Commandino (1506-1575) in 1565 had established the computation of centers of gravity as a problem very much of interest in mathematics, and it is perhaps noteworthy that Commandino himself, who was familiar with the length and breadth of Greek mathematics, wrote only one major original mathematical work, Liber De centro gravitatis solidorum, the focus of which was to determine the center of gravity of a parabolic conoid [3].

Later authors would add tremendously to this body of research on centers of gravity. Among others, Luca Valerio (1552-1618) computed centers of gravity for the "Archimedean Solids" in his work De centro gravitatis solidorum libri tres in 1604 [4]. In 1632, Jean Charles della Faille (1597-1652) devoted an entire work to determining the center of gravity of the sector of a circle [5]. Guldin's 1641 book, usually referred to as the Centrobaryca, was perhaps the most extensive of all of these works, amounting to more than 700 pages devoted solely to the study of centers of gravity. Indeed, the Pappus-Guldin Theorem occupies a small part of the overall work, appearing as Proposition 3 of Chapter Eight of Book II of the Centrobaryca. Unfortunately, it is dressed in technical language which makes it almost incomprehensible to the casual reader.

In fact, neither Pappus nor Guldin has given us a legitimate proof of the Pappus-Guldin Theorem. Pappus' proof is lost and, as H. Bussard notes in Guldin's DSB entry, in the Centrobaryca Guldin "attempted to prove his theorem by metaphysical reasoning" [6]. Indeed, for quite a while, there were serious questions about whether Guldin had actually plagiarized his result from Pappus. (See [7, p. 139].) However, Ivor Bulmer-Thomas has successfully argued that this is not the case [8]. Ironically, given the result's attribution to Pappus and Guldin, the first acceptable published proof was presented by the obscure Italian mathematician John Antonio Roccha in Evangelista Torricelli's Opera Geometrica in 1644. (See also [9, p. 157].)

Appearing in 1668, Gregory's proof was not the first proof of the Pappus-Guldin Theorem, but it is noteworthy in that it introduces a new 3-dimensional geometrical figure called the "trunk" to approach the problem. Given a planar figure there are two seemingly disconnected types of 3-dimensional solids constructed from this figure, right cylinders and solids of revolution.

In our diagrams we use the torus and the circular cylinder, the solid of revolution and cylindrical figure generated from a circular base, because they are easiest to render, but any 2-dimensional figure will suffice as a base. Gregory's trunk is a third type of solid figure constructed from the planar figure that connects the two other solid figures using Euclidean proportion theory.

To construct a trunk, start with a 2-dimensional figure. This figure can be stacked on itself to form a right cylindrical figure in the usual way or it can be revolved around an axis of revolution lying in the same plane as the original 2-dimensional figure to form a solid of revolution. To form a trunk, slice the cylindrical figure by a plane passing through the axis of revolution. The portion of the cylinder lying between the slicing plane and the base plane (the shaded portion of the figure below) is what Gregory calls simply the "trunk" of the cylindrical figure. When he needs to distinguish between the portion of the cylindrical figure above the slicing plane and the portion below the slicing plane he will also refer to the portion above the slice (the wire frame in the figure below) as the "upper trunk" and the portion below the slice as the "lower trunk" of the cylindrical figure.

The top of the cylindrical figure lies in a plane parallel to the plane of the base. The slicing plane which makes the trunk will intersect this in a line l. Through the line l drop a plane perpendicular to the top plane. This plane will also be perpendicular to the base plane and intersect it in another line k parallel to the axis of revolution. Gregory calls the distance between k and the axis of revolution the radius of rotation. Note that there are actually many trunks, depending on the choice of slicing plane. However, we assume that we have chosen one fixed trunk and consequently the radius of rotation will be fixed as well.

The bulk of Gregory's work in proving the Pappus-Guldin theorem will be devoted to establishing proportions between the trunk and the cylindrical figure, as well as between the trunk and the solid of revolution.

Gregory treats separately the ratio between the trunk and the solid of revolution and the ratio between the trunk and the cylinder. The key to understanding the first ratio is Cavalieri's Principle: "If two plane (or solid) figures have equal altitudes, and if sections made by lines (or planes) parallel to the bases and at equal distances from them are always in the same ratio, then the plane (or solid) figures also are in this ratio". (See [10, pp. 315-321], cited in [11, p. 516].) Once Gregory establishes a fixed ratio between corresponding slices of the trunk and the solid of revolution, this principle will imply that the solids themselves have the same ratio. He begins with the trunk, slicing it by an arbitrary plane OPV perpendicular to the axis of rotation.

Let V denote the point of intersection of OPV with the axis of rotation and let O and P denote the intersections of OPV with l and k, respectively. Then

Note that the values of OP and PV will not change no matter what the choice of the perpendicular plane OPV, but the intersection of OPV with the trunk will form a trapezoid EFGH which will vary in size as OPV moves along the axis of rotation. By similar triangles,

\[\frac{OP}{ PV} = \frac{GF}{FV} \]

and

\[\frac{OP}{ PV} = \frac{HE}{EV}\]

Multiply the first equality by \(\frac{1}{2\pi}\). Then some elementary arithmetic shows that

The second equality shows similarly that

Consequently, by applying Euclid V.19 to these two ratios,

where area(GHEF) denotes the area of the trapezoid GHEF and area(annulus(FV - EV)) denotes the area of the annulus obtained by revolving the segment EF around the axis of rotation. Note that the numerator on the right is a slice of the trunk and the denominator is the corresponding slice of the solid of revolution. Since OP and PV do not change, it follows by Cavalieri's principle that the ratio of the volume of the trunk to the volume of the solid of revolution is equal to the ratio of OP to 2\(\pi\)PV.

To be more specific, if AB is any planar figure let rev(AB) denote the volume of the solid of revolution obtained by revolving AB around an axis of revolution and trunk(AB) denote the volume of a trunk of a right cylinder over AB. Then

If OP = height(AB) is the height of the cylinder over AB and \(2 \pi \)PV = circum(AB) is the circumference of the circle with radius equal to the radius of rotation for AB, then we can write

This is the sought after ratio between the solid of revolution and a trunk constructed from the same 2-dimensional figure.

This formula also yields a way to describe the ratio between the volumes of two solids of revolution--something quite important for someone brought up to appreciate Euclidean proportion theory. Suppose, for instance, that AB and EF are two planar figures extended into 3-dimensions to form cylindrical figures which by assumption have the same height.

Using the notation from above, the previous result implies that

Since height(AB) = height(EF) by assumption, we eliminate the common value from both equations to arrive at

This shows that the ratio between solids of revolution can be understood completely in terms of trunks and radii of rotation.

For most mathematicians, the ratio that Gregory establishes between the trunk and the solid of revolution is easier to grasp than the ratio he establishes between the trunk and the cylinder. This stems largely from differences in mathematical upbringing. As noted previously, centers of gravity were an important and familiar part of the mathematical culture during the sixteenth and seventeenth centuries. But apart from the intuitive definition of the center of gravity as the balancing point for a region, they are largely absent from the training of most mathematicians today. However, the most important result for an intuitive understanding of centers of gravity is probably familiar to those who have seen second semester calculus:

Suppose "two masses m₁ and m₂ are attached to a rod of negligible mass on opposite sides of a fulcrum and at distances d₁ and d₂ from the fulcrum. The rod will balance if m₁d₁ = m₂d₂" [12, p. 554]. This is a restatement of something first given as Proposition I.6 of Archimedes' On the Equilibrium of Planes. After some division, it states the result (more amenable to someone versed in proportion theory) that two volumes balance at a point such that the ratio of the distances of this point to the centers of gravity of the volumes is reciprocally proportional to the ratios of the two volumes. That is, m₁/m₂ = d₂ / d₁. In what follows we will occasionally have need to rely on this and other intuitive results on centers of gravity.

Gregory's first task is to locate the center of gravity of the trunk. To start things off, Gregory limits himself to right cylinders constructed from figures that are symmetric about an axis perpendicular to the axis of rotation of the solid of revolution.

This restriction is very useful: If a figure AB is symmetric, its center of gravity will lie somewhere on its axis of symmetry. Furthermore, if a is the center of gravity of AB and b is the center of gravity of the top of the right cylindrical figure built on AB, then the center of gravity of this entire cylindrical figure lies at the midpoint X of the segment ab.

In particular, let IFQS be a rectangle which (1) passes through the axis of symmetry IS, (2) is perpendicular to the axis of rotation, and (3) has the same height FI = SQ as the cylindrical figure. Furthermore, suppose R and H are the respective midpoints of QS and FI. Then the center of gravity X of the cylindrical figure will lie at the intersection of the line segments ab and RH.

All of this can be used to determine the center of gravity of the trunk. Suppose that the (symmetrical) cylindrical figure is sliced by a plane through the axis of rotation to form an upper trunk and a lower trunk as before. For simplicity of notation, assume that the axis of symmetry IS is also a radius of rotation. Reasoning by symmetry again shows that the centers of gravity of both the upper trunk and the lower trunk will lie on the plane IFQS.

Since FI is twice HI, symmetry and Euclid VI.2 together show (with some work) that if the lower trunk is sliced by a plane perpendicular to IFQS to form a rectangular slice KLMN of the trunk, then KLMN will have its geometrical center on the line HS.

Since the geometrical center of a rectangle is also its center of gravity, it follows that the center of gravity of such a slice of the trunk will lie on the line HS. Since this is true for each slice KLMN of the trunk an argument from indivisibles then shows that the center of gravity Y of the lower trunk must lie on the line HS. A similar argument shows that the center of gravity Z of the upper trunk must lie on the line FR.

But where on HS will the lower trunk's center of gravity lie? As Archimedes had demonstrated, if the mass of the upper trunk is concentrated at its center of gravity Z and the mass of the lower trunk is concentrated at its center of gravity Y, then the center of gravity of the upper and lower trunk combined will be a point on the segment YZ such that the ratio of the distance from this point to Y and Z, respectively, will be inversely proportional to the ratio of the volume of the upper trunk to the lower trunk. But the two trunks combined make up the entire cylinder, the center of gravity of which is X.

It follows that X, Y, and Z are collinear and

where volume(upper) and volume(lower) denote the volumes of the upper and lower trunks, respectively.

But X lies on HR, and Z and Y lie on FR and HS, respectively. Also, the lines HR, FQ, and IS are parallel to each other, as are the lines QS, FI, and ab, as well as the lines FR and HS. So ΔHYX is similar to ΔRZX and

The final trick to determining the ratio between the trunk and the cylinder is to use Euclid V.18 (a.k.a. adding "one" to both sides of the above proportion) to arrive at

where cyl(AB) denotes the volume of the cylinder over AB and trunk(AB) denotes the volume of the lower trunk over AB as before. Now IS is the radius of rotation. Also, recall that a is the center of gravity of the original figure. Therefore, aS is the distance between the center of gravity of the original figure and the axis of rotation. So after taking reciprocals and multiplying numerator and denominator by 2π, we have proved the key ratio between the trunk and the cylinder: If AB is a 2-dimensional figure symmetric about an axis, then

where circum(a) denotes the circumference of the circle with radius the distance between the center of gravity and the axis of rotation and circum(AB) denotes the circumference of the circle with radius equal to the radius of rotation.

If AB and EF are two planar figures with centers of gravity a and e, respectively, and which generate cylinders with the same height, we saw previously that

Next, we multiply by one (in the very useful form something/something) twice and substitute in the ratio between the trunk and cylinder from the last section. This gives

where this last equation follows since the cylinders over AB and EF have the same height and hence the volumes are in the same ratios as the bases.

Notice that this is Pappus' version of the Pappus-Guldin theorem. The problem is that this proof only holds if both of the figures AB and EF are symmetric about an axis perpendicular to the axis of rotation. But it is not difficult to drop this requirement. If AB is not symmetric around such an axis, it can be reflected around an axis perpendicular to the axis of rotation to generate a figure HI consisting of two copies of AB which is symmetric around an axis.

Archimedes' result shows that the center of gravity h of HI will be the midpoint of the segment connecting the centers of gravity of the two reflected copies of AB. Since the axis of reflection is perpendicular to the axis of rotation, the centers of gravity of the two reflected copies of AB will be the same distance from the axis of rotation and hence the line connecting them will be parallel to the axis of rotation. In other words, the center of gravity of HI will be the same distance from the axis as the center of gravity a of AB, or circum(a) = circum(h). Note also that area(HI) = 2 area(AB) and rev(HI) = 2 rev(AB) since HI is twice the size of AB. Similar arguments apply to a symmetric figure JK with center of gravity j constructed in the same way from EF. So the result for symmetric figures implies

Cancel off the 2's and the restriction of having a figure symmetric about an axis is removed.

With all of this proportion theory in hand, Gregory's proof of the Pappus-Guldin Theorem falls into place relatively easily. Suppose that AB is the geometrical figure which is to be rotated around an axis and that a is its center of gravity. The central idea of his proof is to use the proportional version of the theorem given in the last section to compare AB with another, easy-to-understand 2-dimensional figure. In this case, that figure is a rectangle HIJK and its axis of rotation is simply the side HI of the rectangle.

For a rectangular figure HIJK, we have area(HIJK) = HI×HK. Since the solid of revolution obtained by revolving HIJK around the the line HI is a cylinder with height HI and radius HK, we get rev(HIJK) = πHI×HK². Finally, the center of gravity h of a rectangle is the geometrical center of the rectangle, so the distance from h to HI is (1/2) HK and thus circum(h) = 2π×(1/2)HK = πHK. With some algebraic simplification, the proportional version of the Pappus-Guldin theorem from the last section then becomes

In particular, the denominators on both sides of the equation are the same. Consequently, the numerators must be equal as well. That is,

which is precisely the Pappus-Guldin theorem.

The reader might be left wondering why we should be interested in James Gregory's proof of this result. Priority--being the first to publish a proof of the result--certainly isn't an issue here. The first published proof of the Pappus-Guldin theorem appeared more than 20 years before Gregory's GPU. It's also not the case that Gregory's proof is more elegant than those presented by his predecessors. Roccha's proof is only about a paragraph long, and once some concepts from the theory of indivisibles are mapped into modern ideas of calculus it does not differ significantly from the proofs found in most calculus books. Perhaps the best apologist for Gregory's work is Gregory himself. Near the end of the GPU he describes his purpose:

These particular problems selected by me, besides certain problems now first solved by me, were found to be very difficult and of great importance among geometers. But indeed Archimedes' entire work "On The Sphere and the Cylinder" is easily demonstrated from Proposition 3 in the manner of Proposition 46 and a few of the following propositions. His book "On Conoids and Spheroids" and all of Luca Valerius' work is demonstrated from Proposition 21; all of the work of Guldin, Ioannis della Faille, and Andreas Tacquet is demonstrated from Proposition 35 and a few of the subsequent propositions.

In other words, Gregory's contribution is abstraction. For Gregory, the Pappus-Guldin Theorem (and quite a few other results) are easy consequences of a broader geometrical perspective--that is, a perspective involving ratios between the trunk, the cylinder, and the solid of revolution.

In making these abstract connections, he is part of a trend that dominated much of 17th century mathematical thinking. Whereas Greek mathematicians and early 17th Century mathematicians had found solutions to specific problems, later mathematicians worked with entire classes of objects and looked at connections between them. As we know, the culmination of this drive to abstraction would be the Fundamental Theorem of Calculus, which would cement the connection between integration and differentiation and almost completely trivialize area and volume computations. For mathematicians today, it is easy to underestimate the revolutionary impact that the Fundamental Theorem of Calculus had on mathematical thinking. But the contrast with the fate of the Pappus-Guldin Theorem helps put it into perspective. As Pappus himself recognized more than 1700 years ago, his theorem gives a truly remarkable and quite surprising connection between volumes of revolution, areas, and centers of gravity. But now it appears essentially as a margin note in calculus texts--completely usurped in its importance by the Fundamental Theorem of Calculus.

So too with James Gregory's contribution to the Pappus-Guldin Theorem. His ratios between right cylindrical figures, solids of revolution, and the trunk give an important and noteworthy connection between the two completely familiar and yet seemingly different ways of constructing solids out of 2-dimensional figures. But the connections and abstractions he saw were geometrical and not analytical. While the fundamental importance of Gregory's "trunk" may seem obvious to someone trained to think in terms of Euclidean proportion theory, what Gregory failed to recognize was that the importance of classical geometry arguments in mathematics was on the decline. Even as Gregory was proving his abstract geometrical connections, Newton and others were already successfully employing the algebraic techniques and methods of analysis that are still in the ascendance in mathematics today.

The author would like to thank Dennis Schneider for his help in using Mathematica to prepare the 3D images used in this article; The author would also like to thank the Centro di Ricerca Matematica Ennio De Giorgi and Pier Daniele Napolitani in particular for their support in the author's participation in the Scientific Revolutions of the XVI and XVII century workshop.

REFERENCES

1. Jones, Alexander. Pappus of Alexandria - Book 7 of the Collection. New York: Springer-Verlag, 1986.
2. Heath, Thomas. The Works of Archimedes. Cambridge: Cambridge University Press, 1897.
3. "Frederico Commandino." Dictionary of Scientific Biography. New York: Scribner, 1970.
4. Napolitani, Pier Daniele and Ken Saito. "Royal Road or Labyrinth? Luca Valerio's De Centro Gravitatis Solidorum and the Beginnings of Modern Mathematics." Bollettino di Storia delle Scienze Matematiche 24 (2004).
5. "Charles de la Faille." Dictionary of Scientific Biography. New York: Scribner, 1970.
6. "Paul Guldin." Dictionary of Scientific Biography. New York: Scribner, 1970.
7. Boyer, Carl. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1959.
8. Bulmer-Thomas, Ivor. "Guldin's Theorem--Or Pappus's?" ISIS 75 (1984) 348-352.
9. Torricelli, Evangelista. Opera di Evangelista Torricelli, vol. 1. Faenza, Italy: 1919.
10. Anderson, Kirsti. "Cavalieri's method of indivisibles." Archive for history of exact sciences 31 (1985) 291-367.
11. Katz, Victor. A History of Mathematics: An Introduction (3rd edition). Boston: Addison-Wesley, 2009.
12. Stewart, James. Calculus: Early Vectors. Pacific Grove, California: Brooks/Cole, 1999.

James Gregory's Proof of the Pappus-Guldin Theorem is given in Proposition 35 of the GPU, but the groundwork for Gregory's proof is carried out in a series of propositions prior to that. A translation of the complete work is available online here. The necessary propositions (Propositions 23,27, 29, 31, 33, and 35) are largely self-contained and translations are included here for completeness. Latin originals of these propositions are given in the next section.

Proposition Twenty-three.

If a right cylindrical figure above a given figure is cut by a plane, the trunk of this cylindrical figure will be to the solid of revolution arising from the rotation of its base around the common intersection of the extended (if necessary) base and the cutting plane as the altitude of the cylindrical figure is to the circumference of the circle whose radius is the radius of rotation.

Let ABDC be a right cylindrical figure above a figure DCF which is cut by a plane KINM in such a way that the figure RGHQ is the common intersection of the plane with the cylinder.

Let the cutting plane be extended until it cuts the base plane DCF in the line KI and the plane of AB in the line MN. From any point, say, L, on the line IK let a plane OLP be drawn perpendicular to IK and cutting the planes IKMN and DFC normally in the lines OL and LP. Let the line OP be perpendicular to LP. We assume that KI is the axis of rotation. We call the line PL normal to this the radius of rotation. I say that the trunk RQDC of the cylinder is to the solid of revolution arising from the rotation of the figure DEFC around the axis of rotation IK as the altitude OP of the cylinder is to the circumference of the circle whose radius is the radius of rotation LP.

Let EF be any line terminated on both sides of the circumference of the base at E and F and drawn through the base DEFC, which, when extended, intersects the axis of rotation normally in the point V. From the points E and F let EH and FG be sent out perpendicular to the base plane and terminated in the cutting plane at H and G. Since the cylinder is right, these are necessarily on the surface of the trunk. Let the line VH be drawn, which is necessarily in the cutting plane IKMN. It is manifest that the triangles OLP and HVE, which have right angles at the points P and E, are similar (since the angles OLP and HVE are equal to the inclination of the cutting plane IKMN). With the line GV having been drawn, the triangles HEV and GFV are similar for the same reason. Since GF is parallel to the line HE and EF intersects EV, the lines GV and HV lie on one line in the plane IKMN. GHEF will be the common intersection of the plane GFV, parallel to the plane OPL, with the truncated cylinder RQDC. Therefore, it is apparent that OP is to PL as HE is to EV. Therefore, as OP is to the circumference of the circle whose radius is PL, thus HE is to the circumference of the circle whose radius is EV. And as OP is to the circumference of the circle whose radius is PL thus the triangle HEV is to the circle whose radius is EV (by drawing the remaining boundaries to the same altitude, namely, half of the line EV). It is demonstrated in the same manner that as OP is to the circumference of the circle whose radius is PL, thus the triangle GFV is to the circle whose radius is FV. Therefore, the entire triangle GFV is to the entire circle with radius FV thus as the cut-off triangle HEV is to the cut-off circle whose radius is EV. Consequently, the remaining trapezoid GFEH will be in the same ratio to the remaining annulus arising from the revolution of the line FE around the axis of rotation IK, namely, in the ratio of OP to the circumference of the circle whose radius is LP. But this proportion is demonstrated in the same manner for all lines drawn in the base DEFC which, when extended (if necessary), meet the axis of rotation IK normally. But the base DC consists of all such lines, the trunk RQDC consists of all of the trapezoids above these lines, and the solid of revolution arising from the revolution of the base around the axis of revolution IK consists of all of the annuli produced by the revolution of these lines. Consequently, as one antecedent is to one consequent -- namely, the altitude OP of the cylinder is to the circumference of the circle whose radius PL is the radius of rotation -- thus, all antecedents -- namely, all trapezoids -- that is, the trunk RQDC -- are to all consequents -- namely, all annuli -- that is, the solid of revolution arising from the rotation of the figure DC around the axis IK, which it was desired to demonstrate.

This theorem is demonstrated in the same manner for the upper trunk if the figure AB is conceived to be rotated around the line MN.

It is apparent from the demonstration that the trunk RQDC and the solid of revolution arising from the revolution of the base DC around the axis of rotation IK are proportional quantities in magnitude and weight, since each proportion which is demonstrated among the entireties is demonstrated in the same manner concerning their proportional parts.

In the following results it ought to be noted that (when we speak about the surface of the cylindrical figure or of the trunk) we understand a surface alone without bases. That is, we never consider figures which are the base of the cylinder, nor do we consider the common intersection of the plane cutting the cylinder.

Proposition Twenty-seven.

If two given right cylindrical figures of equal height are both cut by a plane into two trunks, the proportion of the solid of revolution arising from the rotation of the base of the cylinder around the common intersection of the extended (if necessary) base with the cutting plane to the solid of revolution arising from the same rotation of the base of the other cylinder is compounded directly from the proportion of the radii of rotation and the proportion of the lower trunks of the cylinder.

Let ABCD and NMOYXZ be two right cylindrical figures of equal height with bases DC and XYZ which are cut by planes together into two trunks. Specifically, let ABCD be cut by the plane KHFG into trunks AB43 and 43DC and let NMOYXZ be cut by a plane PSVR into trunks NMO765 and 765ZXY. Let the lines KH and GF be the intersections of the plane HFGK with the planes of the extended (if necessary) parallel bases DC and AB of the cylinder, and let the lines RP and VS be the intersections of the plane RVSR with the planes of the extended (if necessary) parallel bases NMO and ZXY of the cylinders. Let there be planes normal to the lines FG and SV which intersect the cutting planes in the lines IE and QT and the planes of the extended (if necessary) DC and ZXY bases in the lines LE and WT, and let the angles ILE and QWT be right.

It is manifest from Proposition 23, with HFGK and PSVR the assumed cutting planes and GF and VS the axes of rotation, that LE and WT are the radii of rotation. Therefore, I say that the solid of revolution arising from the rotation of the figure DC around GF is to the solid of revolution arising from the rotation of the figure ZXY around VS in the ratio compounded from the proportion of the trunk 34CD to the trunk 567YXZ and from the proportion of the radius of rotation LE to the radius of rotation WT.

The ratio of the solid of revolution arising from the figure DC to the solid of revolution arising from the figure ZXY is compounded from the ratio of the solid of revolution arising from DC to the trunk 34CD, from the ratio of the trunk 34CD to the trunk 567YXZ, and from the ratio of the trunk 567YXZ to the solid of revolution arising from YXZ. But the ratio of the solid of revolution arising from DC to the trunk 34CD is equal to the ratio of the circumference of the circle described by the radius LE to the altitude IL of the cylinder. (See Proposition 23.) The ratio of the trunk 567YXZ to the solid of revolution arising from YXZ is equal to the ratio of the altitude QW -- that is, IL -- of the cylinder to the circumference of the circle described by the radius TW. Consequently, the ratio of the solid of revolution arising from DC to the solid of revolution arising from ZXY is compounded of the ratio of the trunk 34CD to the trunk 567YXZ, from the ratio of the circumference of the circle described by the radius LE to the line IL, and from the ratio of the line IL to the circumference of the circle described by the radius TW. But these last two ratios compound to the ratio of the circumference described by the radius LE to the circumference of the circle described by the radius TW, which is the same as the ratio of the radius LE to the radius TW. Consequently, the solid of revolution arising from the rotation of the figure DC around the axes FG is to the solid of revolution arising from the rotation of the figure XYZ around the axis VS in a ratio compounded from the proportion of the lower trunk 34CD to the lower trunk 567YXZ and from the proportion of the radius of rotation EL to the radius of rotation TW, which ought to have been demonstrated.

Proposition Twenty-nine.

If a right cylindrical figure above a figure symmetric around an axis is envisioned which is cut by a plane into two trunks such that a plane drawn through the axes on opposite bases of the cylinder is normal to that cutting plane, then one trunk will be to the other trunk as the reciprocal of the parts of the radius of rotation having been cut by the center of gravity of the figure.

Let the right cylinder ABDMLK above a figure KLM symmetric around the axis KN be cut into trunks ABDZY2 and ZY2KLM by a plane ETVG normal to the plane ACNK drawn through the axes AC and KN of the opposite bases of the cylinder. Let P and O, which are joined by a line PO, be the centers of gravity of the opposite bases. Let the cutting plane be extended until it intersects the extended (if necessary) axes AC and KN in the points F and S. Let FI and SQ be perpendiculars to those same extended (if necessary) axes. It is manifest that FISQ is a rectangular parallelogram, also that FI is an altitude of the cylinder, and that IS is a radius of rotation which is certainly perpendicular to the intersection of the cutting plane and the base LKM -- that is, to the line TV -- since it is drawn in the plane FQSI, which is normal to both the cutting plane and the base LKM. I say that the trunk ABDZY2 is to the trunk 2YZMLK as the reciprocal of the ratio of IO to OS.

From the midpoints of the lines FI and QS -- namely, H and R -- let the lines HS, RF, and HR be drawn, with HR cutting OP in X. Thus FI, PO, and QS; and likewise FQ, HR, and IS; and likewise FR and HS will be parallel and equal among themselves. Since FR bisects QS, it will also bisect all the lines in the triangle FQS equidistant to QS itself. Consequently, it will bisect all the diameters of the rectangles in the trunk ABDZY2 which are cut normally by the plane FISQ. Therefore, it will pass through the centers of gravity of all of those rectangles.

Since the trunk consists of all of those rectangles, the line FR will pass through the center of gravity of the trunk. Suppose this point is 3. It is demonstrated in the same manner that the center of gravity of the trunk YZMLK2 is on the line HS. Therefore, since the midpoint X of the line OP is the center of gravity of the entire cylinder, if the line 3X4 is drawn through X from 3 until it intersects the line HS in 4, 4 will be the center of gravity of the trunk YZMLK2. Because the triangles XR3 and XH4 are similar on account of RF and HS being parallel, as X4 is to X3 thus XH will be to XR -- that is, IO will be to OS. But as X4 is to X3 thus the trunk ABDZY2 will be to the trunk ZYMKL2. Consequently, as the trunk ABDZY2 is to the trunk YZKMK2 thus IO will be reciprocally to OS, which ought to have been demonstrated.

Consequence.

Consequently, by componendo the entire cylinder ABDMLK is to the lower trunk YZMLK2 as the radius of rotation IS is to the distance between the center of gravity of the original figure and axis of rotation of the original figure--namely, OS.

Proposition Thirty-one.

If there are two figures symmetric around an axis which are rotated in such a way that the axes of rotation of each of the figures are normal to the axis of each figure, then the ratio of one solid arising from such a rotation to the other solid arising from the same rotation is compounded directly from the ratio of the first figure to the other figure and from the ratio of the segment between the center of gravity and the axis of rotation of the first figure to the similar segment of the other figure.

Let ABC and HIL be any two figures symmetric around the axes BF and IN, which are rotated around the lines EG and MO cutting the extended (if necessary) axes BF and IN normally at the points F and N. Let D and K be the centers of gravity of the figures ABC and HIL. I say that the ratio of the solid arising from the figure ABC rotated around the line EG to the solid arising from the figure HIL rotating around the line MO, is compounded from the ratio of the figure ABC to the figure HIL and from the ratio of DF to KN.

Above the figures ABC and HIL let right cylindrical figures of equal height be cut by planes passing through the lines EG and MO, each one into two trunks, namely, an upper and a lower trunk. The ratio of the solid of revolution arising from ABC to the solid of revolution arising from HIL is compounded from the ratio of the lower trunk of the cylinder above ABC to the lower trunk of the cylinder above HIL and from the ratio of the radius of rotation of the figure ABC to the radius of rotation of the figure HIL.

But the lower trunk of the cylinder above ABC is to the lower trunk of the cylinder above HIL is in a ratio compounded from the ratio of the lower trunk of the cylinder above ABC to the entire cylinder above ABC, from the ratio of the entire cylinder above ABC to the entire cylinder above HIL, and from the ratio of the entire cylinder above HIL to its lower trunk. But from the convertendo of the Consequence to Proposition 29 the lower trunk of the cylinder above ABC is to the entire cylinder as FD is to the radius of rotation of the figure AB. Also, the cylinder above ABC is to the cylinder above HIL as the figure ABC is to the figure HIL. Similarly, by the Consequence to Proposition 29, the cylinder above HIL is to its own lower trunk as the radius of rotation of the figure HIL is to KN. Consequently, the ratio of the lower trunk of the cylinder above ABC to the lower trunk of the cylinder above HIL is compounded from the ratio of the line DF to the radius of rotation of the figure ABC, from the ratio of the figure ABC to the figure HIL, and from the ratio of the radius of rotation of the figure HIL to the line KN. Therefore, the ratio of the solid arising from the rotation of the figure ABC to the solid arising from the rotation of the figure HIL is compounded from the ratio of the figure ABC to the figure HIL, from the ratio of the line DF to the radius of rotation of the figure ABC, from the ratio of the radius of rotation of the figure ABC to the radius of rotation of the figure HIL, and from the ratio of the radius of rotation of the figure HIL to the line KN. But the last three ratios compound to the ratio of DF to KN. Therefore, the ratio of the solid arising from the rotation of the figure ABC around EG to the solid arising from the rotation of the figure HIL around MO is compounded from the ratio of the figure ABC to the figure HIL and from the ratio of the segment between the center of gravity of the figure ABC and its axis of rotation -- namely, DF -- to the segment between the center of gravity of the figure HIL and that same axis of rotation -- namely, KN -- which ought to have been demonstrated.

Proposition Thirty-three.

If there are any two figures which are rotated around a given axis, the ratio of the one solid arising from such a rotation to the other solid arising from the same rotation will be compounded from the direct ratio of one figure to the other figure and from the direct ratio of the segment between the center of gravity and the axis of rotation of the one figure to the similar segment of the other figure.

Let ABC and NQP be any two figures which are rotated around the lines EF and Y4, and let their centers of gravity be D and R, which are sent down lines DG and RZ perpendicular to the axes of rotation EF and Y4. I say that the ratio of the solid arising from the figure ABC rotated around the line EF to the solid arising from the figure NQP rotated around the line Y4 is compounded from the ratio of the figure ABC to the figure NQP and from the ratio of DG to RZ.

Let the lines BH and Q2 touching the figures ABC and NQP at B and Q be drawn parallel to the lines DG and RZ. Let the figures ABC and QNP be conceived to be revolved around the lines BH and Q2, like axes, until, attaining the plane on the other part of the axes, they make the figures BLM and QTX, equal and similar to themselves and having exactly the same position toward the lines BH and EF, and Q2 and Y4. Let O and V be the centers of gravity of the figures BLM and QTX. Let the lines OI and V3 be drawn perpendicular to the lines EF and Y4. Also, let the lines DO and RV, intersecting the lines BH and Q2 in the points K and S, be joined.

It is manifest that the point K is the center of gravity of the figure BACBLM symmetric around the axis BH and likewise that the point S is the center of gravity of the entire figure QNPQTX around the axis Q2. It is also apparent that the line DG, KH, OI and also RZ, S2, V3, are equal among themselves. Since the figures BACBLM and QNPQTX are symmetric around the axes BH and Q2, which are normal to the axis of rotation, therefore the solid of revolution arising from the rotation of the figure BACBLM around the line EF is to the solid of revolution arising from the rotation of the figure QNPQTX in the ratio compounded from the ratio of the figure BACBLM to the figure QNPQTX and from the ratio of KH to S2 by Proposition 31. But the solid arising from the figure BACBLM rotated around EF is twice the solid arising from the figure BAC rotated around the same EF. Likewise, the solid arising from the figure QNPQTX rotated around the line Y4 is twice the solid arising from the figure QNP rotated around the same Y4. Also, the figure BACBLM is twice the figure BAC and the figure QNPQTX is twice the figure QNP. Since halves are in the same ratio with their own doubles, the solid of revolution arising from the figure ABC rotated around the line EF will be to the solid of revolution arising from the figure NQP rotated around the line Y4 in the ratio compounded from the ratio of the figure ABC to the figure NQP and from the ratio of KH to S2--or DG to RZ--which it was desired to demonstrate.

Consequence.

It follows that if the centers of gravity of the figures are equally distant from the axes of rotation, the solids of revolution arising from the rotation of figures are in a direct ratio to the figures themselves. If the figures themselves are equal, it follows that the solids of revolution arising from their rotation are in a direct ratio to the segments between the centers of gravity and the axes of rotation. If the segments and figures are equal, the solids of revolution arising from them will be equal even if the figures are dissimilar between themselves.

Scholium.

From these results, it is manifest that between any two figures there are three ratios--namely, of the one figure to the other figure, of the solid of revolution arising from the rotation of one figure to the solid of revolution arising from the rotation of the other figure, and of the segment between the center of gravity and the axis of rotation of the one figure to the similar segment of the second figure--giving two of which always discloses the unknown third.

All these things are demonstrated universally In the same manner for every curve or curves not enclosing a figure thus so of all geometrical demonstrations these are maximally universal.

Proposition Thirty-five.

Each solid of revolution is equal to a right cylindrical figure whose base is the figure out of the rotation of which the solid is produced and whose altitude is the circumference of a circle in which the center of gravity of the figure is revolved.

Let AB be a figure whose center of gravity is C. Let a solid of revolution be made from the rotation of the figure AB around the line DF. I say this solid of revolution is equal to the cylinder whose base is the figure AB and whose altitude is the circumference of the circle in which the center of gravity C is revolved.

Let HGKI be a rectangle whose center of gravity is L.

Let a cylinder be conceived from the rotation of the rectangle HGKI around the side HI. From the centers of gravity C and L let perpendicular lines CE and LO, which are the radii of rotation, be sent down to the axes of rotation DF and HI. It is manifest that the cylinder arising from the rotation of the rectangle HK around HI is equal to the solid whose base is a circle with radius IK and whose altitude is HI -- that is, to the solid whose base is the rectangle on IK and the semicircumference of the circle with radius IK, or the entire circumference of the circle with radius OL, and the altitude HI. This is the same as the solid whose base is the rectangle HK and whose altitude is the circumference of the circle on the radius OL. But the cylinder from the rotation of HK around HI is to the solid of revolution from the rotation of AB around DF in a ratio compounded from the proportion of the figure HK to the figure AB and from the line OL to the line EC -- or the circumference on the radius OL to the circumference on the radius EC. (See Proposition 33.) But the solid whose base is HK and whose altitude is the circumference on the radius OL -- or the cylinder generated by the rotation of the figure HK around HI -- is in the same ratio to the solid whose base is AB and whose altitude is the circumference of the radius EC. Therefore, the cylinder from the rotation of the figure HK around HI is to the solid of revolution from the rotation of the figure AB around DF as the same cylinder is to the solid whose base is AB and whose altitude is the circumference of the circle on the radius EC. Consequently, the solid of revolution from the rotation of the figure AB around DF is equal to the cylinder whose base is AB and whose altitude is the circumference of the radius EC, which it was desired to demonstrate.

The section contains the Latin originals of Propositions 23,27, 29, 31, 33, and 35. The complete work is available online here.

Prop. 23. Theorema.

Si cylindricus rectus existens super qualibet figura, secetur plano quilibet truncus huius cylindrici erit ad solidum rotundum oritum ex eius base rotata circa communem sectionem baseos (si ope est) producta & plani secantis, vt altitudo cylindric ad circus ferentiam circuli, cuius semidiameter est radius rotationis.

Sit cylindricus rectus ABDC super figura quacunque DCF, qui secetur plano quocunque KINM, ita vt communis intersectio plani cum cylindrico siat figura RGHQ.

producatur planum secans dones baseos DCF planum secet in recta KI, & planum AB in recta MN: & puncto quolibet rectae IK nempe L, ducatur eidem IK perpendiculare planum OLP, secans plan IKMN, DFC, normaliter in rectis OL, LP; sitque perpendicularis in LP recta OP. Supponimus hic KI esse axem rotationis; & rectam PL illi normalem, appellamus radium rotationis. dico truncum cylindrici RQDC esse ad solidum rotundum ortum ex rotatione figurae DEFC circa IK axem rotationis, vt OP altitudo cylindrici ad circumferentiam circuli, cuius semidiameter est radius rotationis LP. per basem DEFC ducatur vbilibet recta EF, a baseos circumferentia vtrinque terminata in E & F, quae producta, axi rotationis normaliter incidat in puncto V: a punctis E, F, excitentur perpendiculares baseos plano EH, FG, a plano secante terminatae in H & G, quae necessario sunt in superficie turnci ob cylindricum rectum, ducaturque VH recta, quae necessario existit in IKMN plano secante: manifestum est triangula OLP, HEV, rectangula ad P & E (cum habeamt angulos OLP, HEV, aequales inclinationi plani secantis I K M N) esse similia; & ducta recta GV, ob eandem rationem similia sunt triangula HEV, GFV, cumque GF sit parallella rectae HE, & EF in directum EV; coincident rectae GV, HV, in vnam rectam plani IKMN, eritque GHEF communis intersectio plani GFV, plano OPL parallelli, cum trunco cylindrici RQDC: patet ergo OP ad PL esse, vt HE ad EV; & ideo vt OP ad circumferentiam circuli cuius semidiameter EV; & vt OP ad circumferentiam circuli cuius semidiameter PL, ita (reliquos terminos in eandem altitudinem, nempe semissem rectae EV, ducendo) triangulum HEV ad circulum cuius semidameter EV: eodem modo demonstratur esse, vt OP ad circumferentiam circuli, cuius semdiameter PL, ita triangulum GFV ad circulum, cuius semidiameter FV: est igitur totum triangulum GFV ad totum circulum cuius semidiameter FV, ita ablatum triangulum HEV ad ablatum circulum cuius semidiameter EV; & proinde in eodem ratione erit relictum trapezium GF EH ad relictam armillam circularem genitam ex reuolutione rectae FE circa axem rotationis IK, nempe in ratione OP ad circumferentiam circuli, cuius semidiameter LP: atque haec proportio eodem modo demonstratur de omnibus rectis ductis in base DEFC, quae (si opus est) productae, in axem rotationis IK normaliter incidunt; atque ex omnibus istis rectis constatur ipsa basis DC; ex omnibus trapeziis super istis rectis descriptis constatur truncus RQDC, & ex omnibus armillis abistarum rectarum reuolutione genitis, constatur solidum rotundum genitum a reuolutione baseos circa axem rotationis IK; & proinde vt vna antecedentium ad vnam consequentium, nimirum OP altitudo cylindri ad circumferentiam circuli cuius semidiameter PL radius nempe rotationis, ita omnes antecedentes, nimirum omnia trapezia, hoc est truncus RQDC, ad omnes consequentes, nempe omnes armillas circulares, hoc est, solidum rotundum ortum ex rotatione figurae DC circa axem IK, quod demonstrare oportuit.

Hoc theorema eodem modo demonstratur de trunco superiore, si figura AB concipiatur rotari circa rectam MN.

Patet ex demonstratione truncum RQDC & solidum rotundum ortum ex reuolutione baseos DC circa axem rotationis IK, esse quantitates magnitudine & grauitate analogas, quoniam eadem proportio quae demonstratur inter integras, eodem modo demonstratur de earum partibus proportionalibus.

In sequentibus notandum (quando loquimur d superficie cylindrici vel trunci) nos intelligere solam superficiem sine basibus; hoc est nunquam consideramus figuras quae sunt cylindrici bases, nec communem secionem plani cylindricum secantis.

Prop. 27. Theorema.

Si duo cylindrici recti quicunque aquialis, secentur a planis quibuscunque, vnusquisque in duos truncos; proportio, solidi rotundi orti ex rotatione baseos cylindrici circa communem baseos (si opus est) producta cum Plano secante intersectionem, ad solidum rotundum ortum ex simili alterius cylindrici baseos rotatione, est composita ex directa proportione radiorum rotationis & directa proportione truncorum cylindrici inferiorum. 

Sint duo cylindrici recti aequialti ABCD, NMOYXZ, basibus DC, XYZ, insistentes, a planis intersecti, vnu quisque in duos truncos; nempe cylindricus ABCD sit interfectus a plano KHFG in truncos AB43, 43DC, & cylindricus NMOYXZ a plano PSVR in truncos NMO765, 76ZXY. Sint plani HFGK cum basium planis cylindrici parallellarum (si opus est) productis, DC, OB, interseciones rectae KH, GF; sintque plani PSVR cum planis basium cylindrici parallellarum NMO, ZXY, (si opus est) productis, interseciones, rectae

RP, VS. Sintque plana rectis FG, SV, normalia, quae plana secantia intersecant in rectis IE, QT, & plana basium DC, ZXY, (si opus est) producta in rectis LE&T; sintque anguli ILE, QWT, recti. Manifestum est ex huius 23, positis HFGK, PSVR, planis secantibus & GF, VS, rotationis axibus, LE, WT, esse rotationis radios. Dico igitur solidum rotundum ortum ex rotatione figurae DC circa GF, esse ad solidum rotundum ortum ex rotatione figurae ZXY circa VS, in ratione composita ex proportione trunci 34CD ad truncum 567YXZ & ex proportione radii rotationis LE ad radium rotationis WT. Ratio solidi rotundi orti ex figura DC ad solidum ortum ex figura ZXY, est composita, ex ratione solidi rotundi ex DC orti ad truncum 34CD, ex ratione trunci 567YXZ ad solidum rotundum ex ZXY ortum; sed ratio solidi rotundi ex DC orti ad truncum 34CD est aequalis rationi circumferentiae circuli ex semidiametro LE descripti ed IL altitudinem cylindrici: & ratio trunci 567YXZ ad solidum rotundum ex YXZ ortum est aequalis rationi QW seu IL altitudinis cylindrici ad circumferentiam circuli ex semidiametro TW descripti; & proinde ratio solidi rotundi orti ex D C ad solidum rotundum ortum ex ZXY est composita, ex ratione trunci 34CD ad truncum 567YXZ, ex ratione circumferentiae circuli ex semidiametro LE descripti ad rectam IL, & ex ratione rectae IL ad circumferentiam circuli ex semidiametro TW descripti; sed hae duae postremae rationes componunt rationem circumferentiae circuli ex semidiametro LE descripti ad circumferentiam circuli ex semidiametro TW descripti, quae eadem est cum ratione semidiametri LE ad semidiametrum TW: & proinde solidum rotundum ortum ex rotatione figurae DC circa axem FG est ad solidum rotundum ex rotatione figurae XYZ circa axem VS in ratione composita ex proportione trunci inferioris 34CD ad truncum inferiorem 567 YXZ, & ex proportione radii rotationis E L ad radium rotationis TW, quod demonstrandum erat.

Prop. 29. Theorema.

Si super qualibet figura circa axem intelligatur cylindricus rectus, ita sectus a plano in duos truncos, vt planum per oppositarum cylindrici basium axes ductum, siat plano secanti normale; truncus vnus erit ad truncum alterum reciproce, vt partes radii rotationis resecta a centro grauitatis figura. 

Svper qualibet figura LKM circa axem KN sit cylinricus rectus ABDMLK sectus in truncos ABDZY2, ZY2KLM, a plano ETVG ad planum ACNK, per oppositarum cylindrici basium axes AC, KN, ductum, normali: sint P, O, centra grauitatis basium oppositarum, quae iungantur recta PO. Producatur planum secans, donec axes AC, KN (si opus est) productis intersecet in punctis F, S; & in eosdem axes (si opus est) productos sint perpendiculares FI, SQ; manifestum est FISQ esse parallellogrammum rectangulum, item FI esse cylindrici altitudinem, & IS rotationis radium, quipipe a intersectionem plani secantis & baseos LKM nempe rectam TV est perpendicularis, quoniam ducitur in plano FQSI, quod vtrique plano & secanti & baseos LKM est normale. Dico truncum ABDZY 2 esse ad truncum 2YZMLK vt reciproce IO ad OS. a mediis punctis rectarum FI, QS, nempe H, R, ducantur rectae H S, R F necnon HR secans OP in X; erunt itaque inter se parallellae & aequles FI, PO, QS, item FQ, HR, IS, item FR, HS, Quoniam F R bifariam secat QS, bifariam quoque secabit in triangulo FQS omnes rectas ipsi QS aequidistantes; & prond bifariam secabit omnes diametros rectangulorum in trunco ABDZY2 a plano FI S Q normaliter secatorum; & ideo transibit per centra grauitatis omnium corundem rectangulorum, cumque ipse truncus cnfletur ex omnibus istis rectangulis; idcirco transibit etiam recta F R per centrum grauitatis ipsius trunci, hoc supponatur esse 3: eodem modo

demonstratur in HS esse centrum grauitatis trunci YZMLK2: cum ergo X medium punctum rectae OP sit centrum grauitatis torius cylindrici; si a 3 per X producatur recta 3X4 donec rectam HS intersecet in 4, erit centrum grauitatis trunci YZMLK: & quia triangula XR3, XH4, sunt similia propter parallellas RF, H , et vt X4 ad X3 ita XH ad XR, hoc est IO ad OS; sed vt X4 ad X3 ita truncus ABDZY2 truncum YZKMK2 ita reciproce IO ad OS, quod demonstrandum erat.

Consectarivm.

Et proinde componendo totus cylindricus ABDMLK est ad truncum inferiorem YXMLK2 vt radius rotationis IS ad distantiam inter centrum grauitatis figurae & axem rotationis eiusdem nempe OS.

Prop. 31.Theorema.

Si sint dua figura quaecunque circa axes, quae sic rotentur vt axes rotationis sint figura vniuseuiusque axi normales; ratio vnius solidi orti ex tali rotatione ad aliud solidum ex eadem genitum, componitur ex ratione directa figura ad figuram; & ex ratione directa intercepta inter centrum gravitatis & axem rotationis vnius figura ad similem interceptiam alterius figurae. 

Sint duae figurae quaecunque ABC, HIL, circa axes BF, IN, quae rotentur circa rectas EG, MO, axes figurarum (si opus est) productos BF, IN, normaliter secantes in punctis F, N, sintque figurarum ABC, HIL, centra gravitatis D, K. Dico rationem, solidi orti ex figura ABC rotata circa rectam EG ad solidum ortum ex figura HIL rotato circa rectam MD, componi ex ratione figurae ABC ad figuram HIL & ex ratione DF ad KN. Super figuris ABC, HIL, intelligantur cylindrici recti aequialti secti a planis transeuntibus per EG, MO, rectas, vnusquisque in duos truncos nempe superiotem & inferiorem. Ratio solidi ex ABC orti ad solidum ex HIL ortum, componitur ex ratione trunci inferioris cylindrici super ABC ad truncum inferiorem cylindrici super HIL, & ex ratione radii

rotationis figurae ABC ad radium rotationis figurae HIL; sed truncus inferior cylindrici super ABC est ad truncum inferiorem cylindrici super HIL in ratione composita, ex ratione trunci inferioris cylindrici super ABC ad totum cylindricum super ABC, ex ratione totius cylindrici super ABC ad totum cylindricum super H & ex ratione totius cylindrici super HIL ad truncum sui inferiorem: sed truncus inferior cylindrici super ABC est totum cylindricum, vt FD ad rotationis radium figurae AI ex consectario huius 29 conuertendo, & cylindricus supra ABC est ad cylindricum super HIL vt figura ABC ad figuram HIL, item cylindricus super HIL est ad truncum sui inferiorem vt radius rotationis figurae HIL ad KN, ex consectario huius 29; & proinde ratio trunci inferioris cylindrici super ABC ad truncum inferiorem cylindric super HIL componitur ex ratione rectae DF ad rotationis radium figurae BC, ex ratione figurae ABC, ad figuram HIL, & ex ratione radii rotationis figurae HIL ad rectam KN: & ideo ratio solidi orti ex rotatione figurae ABC ad solidum ortum ex rotatione figurae HIL componitur, ex ratione figurae ABC figuram HIL, ex ratione rectae DF ad radium rotationis figurae ABC, ex ratione radii rotationis figurae ABC ad radii rotationis figurae HIL, & ex ratione radii rotationis figurae HIL ad rectam KN: sed tres postremae rationes componum rationem DF ad KN; & igitur ratio solidi orti ex rotatio figurae ABC circa EG solidum ortum ex rotatione figurae HIL circa MO componitur ex ratione figurae ABC ad figuram HIL, & ex ratione interceptae inter centrum gravitatae figurae ABC & eius axem rotationis, nempe DF, ad interceptam inter centrum gravitatis figurae HIL & eiusdem axem rotationis, nempe KN, quod demonstrare oportuit.

Prop. 33. Theorema.

Si sint duae figurae quaecunque, quae rotentur circa axes quoscunque ratio vnius solidi orti ex tali rotatione ad solidum alterum ex eadem genitum, componetur ex directa ratione figurae ad figuram; & ex directa ratione interceptae inter centrum gravitatis & axem rotationis vnius figurae ad similem interceptam alterius figurae. 

Sint duae figurae quaecunque ABC, NQP, quae rotentur circa rectas EF, Y4: sintque earum contra gravitatis D, R, e quibus in axes rotationis EF, Y4, demittantur rectae perpendiculares DG, RZ. dico rationem solidi orti ex figura ABC rotata circa rectam EF ad solidum ortum ex figura NQP rotata circa rectam Y4 componi ex ratione figurae ABC, ad figuram NQP, & ex ratione DG ad RZ. Parallellae rectis DG, RZ, ducantur rectae BH, Q2, figuras ABC, NOP, tangentes in B, Q: & circa rectas BH, Q2, sicut axes, concipiantur reuolui figurae ABC, QNP, donec ex altera axium parte planum attingentes, efficiant figuras BLM, QTX, sibi ipsis aequales, similes, & ad rectas BH, EF; Q2, Y4, eandem prorsus positionem habentes: sint figurarum BLM, QTX, centra gravitatis, O, V; ducantur in rectas EF, Y4, perpendiculares OI, V3, iungantur quoque rectae DO, RV, rectas BH, Q2, intersecantes inpunctis K, S: manifestum est punctum K esse centrum

gravitatis integrae figurae BACBLM circa axem BH, item punctum S esse centrum gravitatis figurae integrae QNPQTX circa axem Q2; patet quoque rectas DG, KH, OI; item RZ, S2, V3, esse inter se aequales. Quoniam figurae BACBLM, QNPQTX sunt circa axes BH, Q2, axibus rotationis EF, Y4, normales; igitur solidum rotundum ortum ex rotatione figurae BACBLM circa rectam EF est ad solidum rotundum ortum ex rotatione figurae QNPQTX in ratione composita ex ratione figurae BACBLM ad figuram QNPQTX & ex ratione KH ad S2 per huius 3I; sed solidum ortum ex figura BACBLM rotata circa EF est duplum solidi orti ex figura BAC circa eandem EF rotata; item solidum ortum ex figura QNPQTX rotata circa rectam Y4 est duplum solidi orti ex figura QNP rotata circa eandem Y4; figura quoque BACBLM dupla est figurae BAC, & figura QNPQTX figurae QNP; cumque dimidia sint in eadem ratione cum suis duplis; solidum rotundum ortum ex figura ABC rotata circa rectam EF erit ad solidum rotundum ortum ex figura NQP rotata circa rectam Y4 in ratione composita ex ratione figurae ABC ad figuram NQP & ex ratione KH ad S2, sue DG ad RZ, quod demonstrare oportuit.

Consectarivm.

Sint sequitur, si centra gravitatis figurarum a rotationis axibus aequaliter distent, solida rotunda ex figurarum rotatione genita esse in directa ratione ipsarum figurarum, quod si figurae ipsae sint aequales, sequitur solida rotunda ex earum rotatione genita esse in ratione directa interceptarum inter centra gravitatis & rotationis axes: quod si interceptae sint aequales & figurae, etiamsi inter se diffimillimae, solida rotunda ex illis orta aequalia erunt.

Scholivm.

Ex dictis manifestum est inter duas quascunque figuras tres esse rationes, nempe; figurae ad figuram, solidi rotundi ex rotatione vnius figurae geniti, ad solidum rotundum ex rotatione alterius figurae genitum, & interceptae inter centrum gravitatis & axem rotationis vnius figurae ad similem interceptam alterius figurae, e quibus duas datas tertiam ignotam semper patefacere.

Prop. 35. Theorema.

Ciune solidum rotundum aequale est cylindrico rectae cuius basis est figura ex cuius rotatione gignitur solidum & altitudo circumferentia circuli in quae circumuoluitur centrum gravitatis figurae.

Sit figura quaecunque AB cuius gravitatis centrum C; ex figurae AB rotatione circa rectam DF fiat solidum rotundum, quod dico esse aequale cylindrico cuius basis AB figura & altitudo circumferentia circuli, in qua circumrotatur centrum gravitatis C. Sit rectangulum HGKI cuius centrum

gravitatis L; ex rotatione rectanguli HGKI circa latus HI concipiatur fieri cylindrus, & ex gravitatum centris C, L, in rotationis axes DF, HI, demittantur perpendiculares rectae CE, LO, quae sunt rotationis radii: manifestum est cylindrum genitum ex rotatione rectanguli HK circa HI aequalem esse solido cuius basis est circulus ex radio IK & altitudo HI, hoc est solido cuius basis est rectangulum ex IK in semicircumferentiam circuli ex radio IK vel totam circumferentiam ex radio OL & altitudo HI, quod idem circum solido cuius basis est rectangulum HK & altitudo circumferentia circuli ex radio OL; sed cylindrus ex rotatione HK circa HI est ad solidum rotundum ex rotatione AB circa DF in ratione composita ex proportione figurae HK ad figuram AB & rectae OL ad rectam EC seu circumferentia radii OL ad circumferentiam radii EC, atque solidum cuius basis HK & altitudo circumferentia radii OL, seu cylindrus genitus ex rotatione figurae HK circa HI, est ad solidum cuius basis AB & altitudo circumferentia radii EC in eadem ratione; & ideo cylinrus ex rotatione figurae HK circal DF vt idem praedictus cylindrus ad solidum cuius basis AB & altitudo circumferentia circuli ex radio EC; & proinde solidum rotundum ex rotatione figurae AB circa DF aequale est cylindrico cuius basis AB & altitudo circumferentia radii EC quod demonstrare oportuit.