What is the point?

‘As in Geometry, the most natural way of beginning is from a Mathematical point … closing’. So begins Robert Hooke’s extended verbal and visual pun on Euclid’s Elements, at the start of his Micrographia (1665). The Elements opens with the definitions of the point and line as Platonic ideals; Hooke begins with the point of a needle and the edge of a razor (to represent the line). Hooke shows how imperfect these physical forms are, even questioning the value of ‘demonstrations made only by the productions of the Ruler and Compasses’. In ‘Observation I. Of the Point of a sharp small Needle’ in the Micrographia Hooke is at his best and it is as enjoyable reading the text as studying the image, which can be seen in high resolution in the scan from the Library of Congress copy. Hooke also examines a printer’s point through the microscope – looking ‘like a great splatch of London dirt’ – and the engraved lines of his own images, drawing attention to the illusionistic nature of the printmaker’s art.

Hooke beings with the point of a needle, standing in for Euclid’s point, defined as ‘that which hath no part’ saying that is it ‘made so sharp, that the naked eye cannot distingish any parts of it’. His microscope shows that far from the smooth cone ending in a point which examination with the naked eye suggests, the surface if rough and tip is flattened not pointed. In a typical Hooke analogy, the point seems broad enough to accommodate a hundred mites with enough room that they would not break their necks by being pushed off the side. In his second observation, illustrated on the same plate, Hooke uses the edge of a razor for the straight line, which in Euclid’s definition is ‘a line which lies evenly with the points on itself’, that is with no wiggles in it. Once again the microscope shows that the apparently straight edge is not so straight after all. Hooke’s purpose in this first Observation is to demonstrate how imperfect the works of man are, compared with the perfection of the works of nature which will be the subject of most of the rest of the book. Going back to the article on the point, Hooke’s engagement with print comes to the fore in his next analogy (after the needle’s point), the printer’s point or full stop.

… I observed many both printed ones and written; and among multitudes I found few of them more round or regular then this which I have delineated in the third figure of the second Scheme, but very many abundantly more disfigur’d; and for the most part if they seem’d equally round to the eye, I found those points that had been made by a Copper-plate, and Roll-press, to be as misshapen as those which had been made with Types, the most curious and smoothly engraven strokes and points, looking but as so many furrows and holes, and their printed impressions, but like smutty daubings on a matt or uneven floor with a blunt extinguisht brand or stick’s end.

Hooke, Micrographia 1665, p. 3.

Hooke Micrographia Scheme 35

Although Hooke finds irregularities in the printed impressions of both relief printed type and intaglio engravings printed on a rolling press, he draws attention to the smoothness of engraved lines. It is this feature of engraving that is exploited to such effect in subsequent plates. The engraved line made with a burin prints a hard smooth curve ending in a fine point. In Scheme 35 this can be seen in the spines and claw on the louse’s leg gripping what is no doubt one of Hooke’s own head hairs. (By the way, Hooke’s verbal description of the louse is hilarious). We are asked to forget that the engraved line is not smooth and sharp and allow our imagination to see the image as the perfect form of nature’s creation. The engraving is, in other words, illusionistic. The use of the imperfect line to represent an ideal form takes us back to Hooke’s warning,

The Points of Pins are yet more blunt [than the points of needles], and the Points of the most curious Mathematical Instruments do very seldome arrive at so great a sharpness; how much therefore can be built upon demonstrations made onely by the productions of the Ruller and Comapsses, he will be better able to consider that shall but view those points and lines with a Microscope.

Hooke, Micrographia (1665), p. 2.

Hooke would not have been under any misconceptions about the nature of geometric diagrams. By analogy with the illusion of the engraved line representing the perfection of nature, the diagram functions symbolically. If you construct an equilateral triangle in the intersections of two circles, the proof that the sides are of equal length is not that they have the same measurement on your drawing. The proof comes from logical reasoning based on ideal points, straight lines and circles. The construction of the diagram only helps to visualise the construction, from which the theorem is demonstrated: QED. What then can Hooke mean when he questions what can be built upon demonstrations made with ruler and compasses?

One interpretation might be that Hooke is disingenuously misunderstanding the abstract nature of a Euclidian diagram. Is he saying that because the drawn diagram is an imperfect representation of a geometric diagram, it is an unreliable aid to the thought processes involved in the proof? Hooke is always playing games with us and in his use of the word – as in a geometrical proof – ‘demonstrations’ he seems hint at this. However, he is also making the pragmatic point that the natural philosopher or architect will be wary of building up a design with ruler and compasses, having looked through the microscope and realising how imprecise the drawing really is. I have discussed this point with Megan McNamee who put it to me that, ‘ … his claim might be that more precise pictures would lead to even subtler insights in mathematics, engineering, natural philosophy, architecture, etc. (which has, arguably, proven true).’

On the other hand, the architect or engineer needs to work like the geometer, that is to use a drawing as an idealised representation. Like the geometrical diagram, it is an aid to thinking and to communication. The builder takes dimensions off a plan with dividers at his peril, he uses the written dimensions. The accuracy of construction does not depend on the accuracy of the drawing. Yet, as Megan says, the finer the resolution of the drawing, the more complex systems can be designed or investigated.

Hooke’s meditation on the nature of the point and the line, the nature of compass and ruler constructions, and the engraved line thus has profound implications for scientific and technical illustration.