Archimedes Quotes
130 Archimedes Quotes
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Let us assume that a fluid has the property that if its parts lie evenly and are continuous, the part which is less compressed is expelled by that which is more compressed, and each of its parts is compressed by the fluid above it perpendicularly, unless the fluid is shut up in something and compressed by something else.
Archimedes
Let us assume that anybody which is borne upwards in water is carried along the perpendicular [to the surface] which passes through the centre of gravity of the body.
Archimedes
That any pyramid is one third part of the prism which has the same base as they pyramid and equal height.
Archimedes
That any cone is one third part of the cylinder which has the same base as the cone and equal height.
Archimedes
Cones of equal height are in the ratio of their bases, and conversely.
Archimedes
If a cylinder be divided by a plane parallel to the base, cylinder is to cylinder as axis to axis.
Archimedes
Cones which have the same bases as cylinders and equal height with them are to one another as the cylinders.
Archimedes
The bases of equal cones are reciprocally proportional to their heights, and conversely.
Archimedes
Cones the diameters of whose bases have the same ratio as their axes are in the triplicate ratio of the diameters of their bases.
Archimedes
Spheres have to one another the triplicate ratio of their diameters.
Archimedes
In a paraboloid of revolution any plane section parallel to the axis is a parabola equal to the generating parabola.
Archimedes
In a hyperboloid of revolution any plane section parallel to the axis is a parabola equal to the generating parabola.
Archimedes
In a hyperboloid of revolution a plane section through the vertex of the enveloping cone is a hyperbola which is not similar to the generating hyperbola.
Archimedes
In any spheroid a plane section parallel to the axis is an ellipse similar to the generating ellipse.
Archimedes
If A be any point on a circle and BC any diameter, it is possible to draw through A a straight line, meeting the circle again in P and BC produced in R, such that the intercept PR is equal to any given length.
Archimedes
Given any two straight lines forming an angle and any fixed point which is not on either line, it is required to draw through the fixed point on a straight line such that the portion of it intercepted between the fixed lines is equal to a given length.
Archimedes
The angular points of the polygon will move along the circumferences of circles.
Archimedes
If a straight line in a plane turn[s] uniformly about one extremity which remains fixed, and return to the position from which it started and if, at the same time as the line is revolving, a point move at a uniform rate along the line starting from the fixed extremity, the point will describe a spiral in the plane.
Archimedes
There are in a plane certain terminated bent lines, which either lie wholly on the same side of the straight lines joining their extremities, or have no part of them on the other side.
Archimedes
I apply the term concave in the same direction to a line such that, if any two points on it are taken, either all the straight lines connecting the points fall on the same side of the line, or some fall on one and the same side while others fall on the line itself, but none on the other side.
Archimedes
Similarly also there are certain terminated surfaces, not themselves being in a plane but having their extremities in a plane, and such that they will either be wholly on the same side of the plane containing their extremities, or have no part of them on the other side.
Archimedes
I apply the term concave in the same direction to surfaces such that, if any two points on them are taken, the straight lines connecting the points either all fall on the same side of the surface, or some fall on one and the same side of it while some fall upon it, but none on the other side.
Archimedes
I use the term solid sector, when a cone cuts a sphere, and has its apex at the centre of the sphere, to denote the figure comprehended by the surface of the cone and the surface of the sphere included within the cone.
Archimedes
I apply the term solid rhombus, when two cones with the same base have their apices on opposite sides of the plane of the base in such a position that their axes lie in a straight line, to denote the solid figure made up of both the cones.
Touching, without cutting.
Archimedes
Of all the lines which have the same extremities the straight line is the least.
Archimedes
Cones having equal height have the same ratio as their bases; and those having equal bases have the same ratio as their heights.
Archimedes
The surface of any sphere is equal to four times the greatest circle in it.
Archimedes
Of all segments of spheres which have equal surfaces the hemisphere is the greatest in volume.
Archimedes
The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumstance, of the circle.
Archimedes
The area of any ellipse is to that of the auxiliary circle as the minor axis to the major.
Archimedes
The areas of ellipses are as the rectangles under their axes.
Archimedes
When it was once proved that the surface of any sphere is four times the greatest circle in the sphere, it is clear that it is possible to find a plane area equal to the surface area of the sphere.
Archimedes
If a straight line touch[s] the spiral, it will touch it in one point only.
Archimedes
Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.
Archimedes
If when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made.
Archimedes
Weights which balance at equal distances are equal.
Archimedes
Unequal weights at equal distances will not balance but will incline towards the greater weight.
Archimedes
Unequal weights will balance at unequal distances, the greater weight being at a lesser distance.
Archimedes
If two equal weights have not the same centre of gravity the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity.
Archimedes
A solid lighter than a fluid will, if immersed in it, not be completely submerged, but part of it will project above the surface.
Archimedes
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