Tangent Quotes (5 quotes)

Curves that have no tangents are the rule.
Those who hear of curves without tangents, or of functions without derivatives, often think at first that Nature presents no such complications.
The contrary however is true.
Consider, for instance, one of the white flakes that are obtained by salting a solution of soap. At a distance its contour may appear sharply defined, but as we draw nearer its sharpness disappears. The eye can no longer draw a tangent at any point.
The use of a magnifying glass or microscope leaves us just as uncertain, for fresh irregularities appear every time we increase the magnification.
An essential characteristic of our flake
is that we suspect
that any scale involves details that absolutely prohibit the fixing of a tangent.

In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.

Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures.

Stay your rude steps, or eer your feet invade

The Muses haunts,ye sons of War and Trade!

Nor you, ye legion fiends of Church and Law,

Pollute these pages with unhallowd paw!

Debased, corrupted, grovelling, and confind,

No definitions touch your senseless mind;

To you no Postulates prefer their claim,

No ardent Axioms your dull souls inflame;

For you no Tangents touch, no Angles meet,

No Circles join in osculation sweet!

The Muses haunts,ye sons of War and Trade!

Nor you, ye legion fiends of Church and Law,

Pollute these pages with unhallowd paw!

Debased, corrupted, grovelling, and confind,

No definitions touch your senseless mind;

To you no Postulates prefer their claim,

No ardent Axioms your dull souls inflame;

For you no Tangents touch, no Angles meet,

No Circles join in osculation sweet!

Too many heretics are simply witch-hunters off on an unaccustomed tangent.