“It may be noted that in Neo-Plastic art the crucial thing is the right angel, that is, the right-angle lines. And not whether the position of the lines is vertical or horizontal... So it is possible to make very beautiful things while placing the lines in a diagonal position.”

— Piet Mondrian

In 'Neo-Plasticism: Home – Street – City', Piet Mondriaan, 'i 10', Jan. 1927
1920's

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Piet Mondrian 95

Peintre Néerlandais 1872–1944

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“If the masc. [masculine] is the vertic. [vertical] line, then a man will recognize this element in the rising line of a forest; in the horizont. [horizontal] lines of the sea he will see his complement. Woman, with the horizont. line as element, sees herself in the recumbent lines of the sea, and her complement in the vert. lines of the forest.”

Piet Mondrian (1872–1944) Peintre Néerlandais

[on his two paintings 'Sea' and ' Trees', both made in 1912 https://upload.wikimedia.org/wikipedia/commons/6/65/Trees%2C_1912%2C_Mondrian.jpg
note in his sketchbook, undated but c. 1912; as quoted in Mondrian, - The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 70
1910's

“With regard to the diagonal, too, I am in complete agreement with you [with Theo van Doesburg ]. As soon as it appears together with straight [horizontal and vertical] lines, I believe it should be condemned... A while back I started a thing entirely in diamonds [diamond-shape] like this [his sketch in the letter of several diamond-forms]. I have to find out if it's possible: intellectually I'm inclined to say it is. There's something to be said for the idea, because perpendicular and flat lines can be seen everywhere in nature; by using a diagonal line I would be canceling that out. But I'm inclined to say that this cannot be combined with perpendicular and flat lines or with different kinds of slanting lines.”

Piet Mondrian (1872–1944) Peintre Néerlandais

quote about the growing controversy between Mondrian and Van Doesburg. concerning the use of diagonal lines
Source: quote from a letter of Mondrian to Theo van Doesburg, undated, c. May 1918; as cited in Mondrian, -The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 120

“If two right lines cut one another, they will form the angles at the vertex equal….
This… is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales…”

Proclus (412–485) Greek philosopher

Proposition XV. Thereom VIII.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“As we all know, the singularity in the art of Bach is the fusion of both levels and lines, the horizontal and vertical line. It’s a real wonder to see that the creation and forming of the horizontal line, the polyphonic structure, also results in this perfect, beautiful vertical line, the harmonic line. As we also know, Bach already used the full harmonic range and radius as no composer before him. My artistic aim of course is to point out the horizontal line in soloistic manner in a dynamically elastic way, but in the same breath to form the harmonic line in a bright field of color (I would call it “harmonic articulation”), to achieve a particular atmosphere of emotions and moods, drama, velocity, vividness, and so on. As we can imagine, these are high demands …”

Burkard Schliessmann classical pianist

Talkings on Bach

“To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle.
According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient.”

Proclus (412–485) Greek philosopher

And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application...
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“In his later works Doesburg tried to destroy static expression by diagonal position of his lines. But in this way the feeling of physic equilibrium which is necessary to enjoy a work of art is lost.”

Piet Mondrian (1872–1944) Peintre Néerlandais

Quote in a letter of Mondrian to Sweeney, 24 May 1943; as cited in: - 102 - Two autobiographical texts (24 May 1943) http://mondrianwritings.nl/wp-content/uploads/2017/03/102.-Two-autobiographical-texts-24-May-1943.pdf
This idea was partly the reason of their mutual split in 1924; in 1929 they reconciled in Paris.
1940's

“I believe that it is possible by means of horizontal and vertical lines, constructed 'consciously' but not 'calculating', guided by a higher intuition and brought to harmony and rhythm – I believe that these fundamental aesthetic shapes – where necessary supplemented by lines in other directions or curved lines, make it possible to arrive at a work of art which is as strong as it is true. For anyone who sees more deeply, there is nothing vague about this; it is only vague for the superficial observer of nature. And 'chance' must be as far removed as 'calculation'. And for the rest it seems to me that it is necessary to keep breaking off the horizontal or vertical line: for if these directions were not countered by others, they would themselves come to signify something 'specific' and thus human.”

Piet Mondrian (1872–1944) Peintre Néerlandais

Quote of Mondrian in a letter to H. P. Bremmer, Paris 29 January 1914; ; as cited in Mondrian, - The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 81
1910's

“If two right lines cut one another, they will form the angles at the vertex equal.”

Proclus (412–485) Greek philosopher

...
This... is what the the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales...
Proposition XV. Thereom VIII.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“Staff standards are not theological abstractions, but scientific approximations, and are evolved for the use of the line, the sole justification of the standards being that they will make line work more efficient. Staff standards being for the benefit of the line and often entrusted to line officials, must be put in the form of permanent instructions so that all may understand what is being aimed at, and deviations by the line be noted and reprimanded.”

Harrington Emerson (1853–1931) American efficiency engineer and business theorist

Source: Efficiency as a Basis for Operation and Wages, p. 70; Cited in: Morgen Witzel (2003) Fifty Key Figures in Management. p. 79

“Gaiety of tone is the luminous dominant, of tint, the warm dominant, of line, lines above the horizontal
Calmness of tone is the equality of dark and light; of tint, of warm and cool, and the horizontal for line.
Sadness of tone is the dark dominant; of tint, the cool dominant, and of line, downward directions.”

Georges Seurat (1859–1891) French painter

Quotes, 1881 - 1890, Letter to Maurice Beaubourg', August 1890

“It may be noted that in Neo-Plastic art the crucial thing is the right angel, that is, the right-angle lines. And not whether the position of the lines is vertical or horizontal... So it is possible to make very beautiful things while placing the lines in a diagonal position.”

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Piet Mondrian 95

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