• Archimedes, On the Equilibrium of Planes
• with the notes of Eutocius (Volume 3 of Heiberg's edition, pp. 264-318)

Centers of Equal-Inclinations of Planes or of Weights of Planes, 1st

Principles

1. We postulate that equal weights from equal lengths incline equally and that equal weights from unequal lengths do not incline equally but incline towards the weight of the larger length.

2. that if when weights incline-equally from certain lengths something is added to one of the weights they do not incline-equally but incline towards that weight to which there was added

3. similarly that if something is taken away from one of the weights they will also not incline-equally, but will incline towards the weight from which it was not taken away.

4. When equal and similar plane figures fit on one another the centers of weights also fit on one another.

5. The centers of weights of unequal but similar figures will be similarly positioned. We say that the points are similarly positioned in similar figures from which straight lines drawn to the equal angles make equal angles to the corresponding sides.

6. If magnitudes incline equally from certain lengths, those equal to them will also incline equally from the same lengths.

7. In every figure whose perimeter is concave-on-the-same-sides, the center of weight must be within the figure. When these are supposed:

Propositions

1. Weights that incline-equally from equal lengths are equal.

2. Unequal weights from equal lengths do not incline-equally, but will incline towards the larger.

3. Unequal weights will incline-equally from unequal lengths and the larger weight from the smaller length.

4. If two equal magnitudes do not have the same center of weight, the center of weight of the magnitude composed from them both will be the middle of the straight-line joining the centers of weight of the magnitudes.

5. If the centers of weight of three magnitudes are positioned on a straight-line, and the magnitudes have equal weight, and the straight-lines between the centers are equal, the center of weight of the magnitude composed from all the magnitudes will be the point which is also the same as the center of weight of the middle magnitude.

Corollary 1: From these it is in fact obvious that in the case of the centers of weight of however many odd magnitudes in multitude which are positioned on a straight-line, if the magnitudes equally distant from the middle one hace equal weight, and the striaght-lines between their centers are equal, the center of weight of the magnitude composed from all the magnitudes will be the point which is also the center of the magnitude in the middle of them.

Corollary 2: If there are magnitudes that are even in multitude and the centers of their weight are positioned on a straight-line, and the middles one of them and those equally distant from them have equal weight, and the straight-lines between the centers are equal, the center of weight of the magnitude composed from all the magnitudes will be the middle of the line joining the centers of weight of the magnitudes, as has been inscribed below.

6. Commensurable magnitudes incline-equally when they have the same ratio for their lengths conversely as their weights.

7. And then even if the magnitudes are incommensurable, they will similarly incline-equally when they have the same ratio from their lengths conversely as their magnitudes.

8. If from some magnitude some magnitude is taken away that does not have the same center as the whole, with the straight-line that joins the centers of weight of the whole magnitude and the one taken way extended on the same side as the center of weight of the whole magnitude and with something taken away from the extended line that joins the mentioned centers, so that it has the same ratio to the line between the centers which the weight of the taken away magnitude has to the weight of the remaining, the center of weight of the remaining magnitude is the limit of the taken away.

Corollary. If some magnitude is added to some magnitude, the center of weight will lie on the line between the centers of weight so that the distance from the center of the whole to the centers of the parts will be inversely proportional to the magnitudes.

9. The center of weight of every parallelogram is on the straight-line that joins the bisections of the opposite sides of the parallelogram.

10. The center of weight of every parallelogram is the point at which the diameters meet.

11. If two triangles are similar to one another and points are similarly placed in relation to the triangles and one point is the center of weight of the triangle in which it is, then the other point is the center of weight of the triangle in which it is.

12. If two triangles are similar, but the center of weight of one of them is on a straight-line which is drawn from some angle to the middle of the base, then the center of weight of the remaining triangle will be on the line drawn similarly.

13. The center of weight of every triangle is on the straight-line which is drawn from the angle to the middle of the base.

14. In every triangle, the point at which the straight-lines drawn from the angles to the middles of the sides of the triangle meet is the center of weight.

15. In every trapezium having two sides parallel to one antoher, the center of weight is on the straight-line which joins the bisections of the parallels, which is divided in such a way that the segment of it having as limit the bisection of the smaller of the parallels to the remaining segment has this ratio, what the line equal to double the larger with the smaller together has the double of the smaller with the larger of the parallels.

Of Equal-incliners, 2nd

1. If two areas are enclosed by a straight-line and the section of a right-angled cone, which we are able to apply to the given straight-line, do not have the same center of weight, the center of weight of the magnitude composed from both of them will be on the straight-line that joins the center of their weight which divides the mentioned straight-line in a way that the segments of them have the same ratio inversely with the areas.

2. If a triangle is inscribed in a segment enclosed by a straight-line and the section of a right angled cone, that has the same base as the segment and an equal height, and again in the remaining segments triangles are inscribed, and repeatedly triangles are inscribed in the remaining segments in the same way, let the figure that comes about in the segment be said to be familiarly inscribed. It is obvious that the straight-lines joining the angles of the inscribed figure, those nearest the vertex and those successively, will be parallel to the base and will be bisected by the diameter of the segment, and cut the diameter into ratios of successive odd numbers with one stated for the vertex of the segment. One must prove these things in order.

If a rectilinear-figure is inscribed familiarly in the segment enclosed by a straight-line and the section of a right angled cone, the center of weight of the inscribed figure will be on the diameter of the segment.

3. If two of the similar segments enclosed by a straight-line and the section of a right angled cone are familiarly inscribed in each rectilinear-figure, but the inscribed rectilinear-figures have the their sides equal in multitude to one another, the centers of the weights of the rectilinear-figures cut the diameters of the segments similarly.

4. The center of the weight of every segment enclosed by a straight-line and the section of a right angled cone is on the diameter of the segment.

5. If a rectilinear-figure is familiarly inscribed in a segment enclosed by a straight-line and the section of a right angled cone, the center of the weight of the whole segment is nearer to the vertex of the segment than the center of the inscribed rectilinear-figure.

6. Given a segment enclosed by a straight-line and the section of a right angled cone it is possible to inscribe familiarly into the segment a rectilinear-figure so that the straight-line between the centers of weight of the segment and the inscribed rectilinear-figure is less than any proposed straight-line.

7. The centers of the weights of two similar segments enclosed by a straight-line and the section of a right angled cone cut the diameters in the same ratio.

8. The center of the weight of every segment enclosed by a straight-line and the section of a right angled cone divides the diameter of the segment to that the part of it at the vertex of the segment is half-again that at the base.

9. If four lines are proportional in continuous proportion, and something is taken that has this ratio to three fifth-parts of the excess by which the largest of the proportionals exceeds the third: what the least has to the excess by which the largest exceeds the least, and something is taken that has this ratio to the excess by which the largest of the proportionals exceeds the third: the ratio that the line equal to the line double the largest of the proportionals and the line four-times the second and the line six-times the third and the line three-times the fourth to the line equal to the line five-times the largest and the line ten-times the second and the line ten-times the third and the line five-times the fourth, then the taken line together will be two- fifth-parts the largest.

9. in the paraphrase of Eutocius

10. The center of weight of every piece taken away from the section of a right angled cone is on the straight-line which is the diameter of the piece and positioned in this manner: when the straight-line is divided into five equals it is on the middle fifth part so that so that its segment nearer to the smaller base of the piece to the remaining piece has the same ratio which the solid that has has its base the square from the larger of the bases of the piece and the height the line equal to double the smaller base and the height together to the solid having its base as the square from the smaller of the bases of the piece and height the line equal to the double of the larger and the smaller of them together.
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