Remembering... that Eratosthenes of Cyrene, employing mathematical theories and **geometrical** methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.

— Chapter VI, Sec. 9

For a daily Agnihotra ( Balivaishva ) a small copper pot is used. The inverted pyramid shaped Agnihotra pot receives, generates, and decentralizes electrosphere ....Apart from the pyramid shape, some other special symmetric **geometrical** designs are also used according to the kind of energy fields and the cosmic currents required to be generated by the Yagna.

— In P.9 (The Integrated Science of Yagna)

Population, when unchecked, increases in **geometrical** ratio. Subsistence
only increases in arithmetical ratio.

— 1798 An Essay on the Principle of Population.

In this science the illustrations and examples are not confined in their effect merely to the practice they afford in the analytical art, but [...] they also store the mind with independent **geometrical** and physical knowledge. Besides, it should be considered, that the only effectual method of impressing abstract formulae and rules upon the memory, and, indeed, of making them fully and clearly apprehended by the understanding, is by examples of their practical application.

— Dionysius Lardner (1823).

Population, when unchecked, increases in a **geometrical** ratio, Subsistence, increases only in an arithmetical ratio.

— Chapter I, paragraph 18, lines 1-2

We may, I believe, anticipate that the chemist of the future who is interested in the structure of proteins, nucleic acids, polysaccharides, and other complex substances with high molecular weight will come to rely upon a new structural chemistry, involving precise **geometrical** relationships among the atoms in the molecules and the rigorous application of the new structural principles, and that great progress will be made, through this technique, in the attack, by chemical methods, on the problems of biology and medicine.

— Nobel Lecture (11 December 1954)

This absence of similarity among military questions naturally brings out the inability of memory to solve them; also the sterility of invariable forms, such as figures, **geometrical** drawings ( épures ), plans ( schémas ), etc. One only right solution imposes itself:;: namely, the application, varying according to circumstances, of fixed principles.

— p. 154 (Precepts and Judgments (1919))

There is still a difference between something and nothing, but it is purely **geometrical** and there is nothing behind the geometry.

—

Both a priori reasoning and experience teach us that as as these funds grow larger the **geometrical** rate of growth by compound interest ultimately defeats itself.

— Part II, Chapter VIII,Ultimate Uses of the Stored Units, p. 103

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity . The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps **geometrical** figures or drawings.

— "Systems of Logic Based on Ordinals," section 11: The purpose of ordinal logics (1938), published in

In a footnote to the first sentence, Turing added: "We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."

Geometry... is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet. By means of optics... the light in buildings can be drawn from fixed quarters of the sky. ...Difficult questions involving symmetry are solved by means of **geometrical** theories and methods.

— Chapter I, Sec. 4

Population, when unchecked, increases in a **geometrical** ratio, Subsistence, increases only in an arithmetical ratio.

— Chapter I, paragraph 18, lines 1-2 (An Essay on The Principle of Population (First Edition 1798, unrevised))

Population, when unchecked, increases in a **geometrical** ratio, Subsistence, increases only in an arithmetical ratio.

— Thomas Malthus (1798)

Various relations being established in geometry between lines constituted under given conditions, as parts of **geometrical** figures, if we choose to adopt the idea of expressing these lines by numerical measures , we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their **geometrical** relations however are absolutely general, and do not refer to any such distinction.

— Rev. Baden Powell,

Various relations being established in geometry between lines constituted under given conditions, as parts of **geometrical** figures, if we choose to adopt the idea of expressing these lines by numerical measures , we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their **geometrical** relations however are absolutely general, and do not refer to any such distinction.

— Rev. Baden Powell,

The nature of the pigments provides the basis for sensations of light and color; that is, brightness, hue and saturation. The **geometrical** demarcation of these qualities provide the physical basis for perception of areas and their shapes. Altogether, these factors constitute the vocabulary of the language of vision, and are acting as the optical forces of attraction.

— György Kepes (1944/1995)

Anaximander relies on the accuracy of geometry in matters beyond the range of any kind of verification in its application to cosmic proportions and also in contradiction to appearance, which suggests that the sun is about as large in diameter as the width of a human foot. The concept of **geometrical** similarity is also the precondition for Anaximander's attempt to construct a map of the world.

— Walter Burkert,

Ptolemy... against the champions of this or that cosmology of the heavens... had dared to claim that it is legitimate to interpret the facts of astronomy by the simplest **geometrical** scheme which will 'save the phenomena,' no matter whose metaphysics might be upset. His conception of the physical structure of the earth, however, prevented him from carrying through in earnest this principle of relativity, as his objections to the hypothesis that the earth moves amply show.

— Edwin Arthur Burtt,

Architecture is an art which is basically **geometrical**. The cube is the basis
of architecture because the right angle is necessarythe
steps of a staircase consist of vertical and horizontal planes
and the corners of rooms are nearly always right
angles.We need right angles. 542

— 1924 'Architecture and Geometry', in Bulletin de la Vie Moderne, Paris.

The mathematical thermology created by Fourier may tempt us to hope that, as he has estimated the temperature of the space in which we move, me may in time ascertain the mean temperature of the heavenly bodies: but I regard this order of facts as for ever excluded from our recognition. We can never learn their internal constitution, nor, in regard to some of them, how heat is absorbed by their atmosphere. We may therefore define Astronomy as the science by which we discover the laws of the **geometrical** and mechanical phenomena presented by the heavenly bodies.

— Book II: Astronomy, Ch. I: General View