I. Dee a representative sixteenth century scientific figure - the endeavour after a new philosophical synthesis (n.i) and neoplatonism - his scientific importance chiefly due to his assumption of the role of a propagandist for mathematics - this championship derived from metaphysical doctrines, not considerations of utility - the essential unity of his thought, though only some discrete elements of it historically survive as significant - and the Elizabethan world order - in what sense Dee is a philosopher - the need for a philosophical defence of mathematics in his time.

II. Impossibility for mere empiricism in sixteenth century to found a successful science - necessity of a preliminary framework of concepts and uniform methodology in any examination of nature - the fact constructed or determined by such contexts - in these the only means to achieve one constant goal of science, to organise its constituent knowledge into a deductive system - such systems differentiated by such contexts - in these the only means to achieve one constant goal of science, to organise its constituent knowledge into a deductive system - such systems differentiated by their fundamental criteria of the significant or intelligible - importance in this respect of the Renaissance for later scientific thought - the subsequent progress of mathematical domination of science first adumbrated then - emergence of fundamental assumptions of Newtonian Science in the Renaissance - the apparent transformation of metaphysics into methodology - ultimate dependence of all practise on theory; and the experimental method - history of science the gradual clarification of ideas not accumulation of facts - contrast between actual development of a science through time and its late formal elaboration - consequent necessity for studying the thought of a period as a "whole" and in its own terms.

III. The main schools of Renaissance metaphysics - inevitable defects of such sketches as present, excessive schematisation, over-simplification, too clear cut distinctions - subsequent chapters correctives to these generalisations - actual picture especially blurred by extensive permeation of naturalism - the plausibility and scientific impotence of this.

IV. The services of the Aristotelian revival of the thirteenth century for later thought - important as preparatory organisation of scientific knowledge - as a system rapidly, and by the Renaissance, fully, exploited - the inadequacy and dangers of its qualitative analysis of nature - its standards of Reality fatally divorced from intelligibility - doctrine of essence and accident and assumption of a direct access to facts through language - ascription of causal efficacy to quiddity - its wide appeal to experience and qualitative "experiment" - inadequacies of Aristotelianism as an overall scheme of correlation - it represented a merely contemplative ordering of knowledge, providing no assistance to applied science - doctrines of causality - species and teleology - discontinuities it established in nature - separation of the realms of faith and reason and consequent low estimate of powers of mind - the rational and the natural and the depreciation of mathematics.

V. The evaluation of mathematics one of principal matters in dispute between Aristotelian and neoplatonist schools in Renaissance - restricted nature of empirical evidence producible against Aristotelianism - but general superiority of mathematical procedures to verbal methods of investigation.

VI. Tradition of scientifically orientated neoplatonism of Dee - its critical but not hostile attitude to Aristotelian doctrines - conciliation of Plato and Aristotle a perennial characteristic of neoplatonism - lack of any differentiation between Plato and his followers - Pythagoreanism.

VII. The constant characteristics of neoplatonism in all its varieties - insistence on the mind's direct access to truth - the status of the universal - ultimate grounds of knowledge placed at opposite end of scale of being to sensible - the Idea of the Good, Union with the One and intellectualism of Christian neoplatonism - the a priori - revelation and the levels of knowledge - thought always directed towards being - transformation of Platonic Ideas into modes of considering objects - and operational conventions - Ideas become essentially functional for Galileo - reconstitution by scientific neoplatonism of Renaissance of "objective" as that of which a logical account can be given.

VIII. The sensible world and neoplatonism - it awakens individual minds to latent conceptual knowledge - its derivation from intellectual truth - its consequent rationality and the new ideal of discoverable natural law - intelligible unity of the cosmos in neoplatonism of Renaissance - universe no longer crudely anthropocentric - the "parts" of the universe - superiority of neo-Platonic mathematical approach over qualitative atomism - the natural fact considered as a theorem in geometry pertaining in the same way to a system - Platonic tradition of the unification of all sciences - and the new stress on detail and exactitude - the revaluation of nature - the unity of the natural and intelligible following on the view of the Real as the logically representable - dialectic and reality in Platonism.

IX. The a priori and mathematics - sixteenth century stress on mathematical doctrine as fundamental distinction between Plato and Aristotle - mathematics for Plato - fusion with Pythagoreanism - the entire certainty of mathematics - mathematics as the type of objectivity - general sixteenth and seventeenth century adoptions of these teachings - Roger Bacon's ideal of "verifying" all the sciences by mathematical means and John Dee.

X. The connections between neoplatonism and new science in sixteenth century - structure gradually replaces tendency as key concept in analysis of nature - the reconciliation of belief in overall Divine Providence and acceptance of universal mechanical causality - inspiration given by the Timaeus - superiority of new mathematical methodology over Aristotelian induction - beneficial results of spread of Ramus' logic directed towards psychology rather than "substance" - its connection with mathematical approach to nature and general advantages of this - idealism of relation replaces realism of substance - germs of subsequent science clearly present in Renaissance - Galileo's neo-Platonic mathematicism - his confidence in reason even when opposed by experience - the scientific inspiration of Archimedes' works in sixteenth century - effect of his apparent success in applying purely a priori methods to establish synthetic results.

XI. The replacement of quality by quantity as what is taken as fundamental in nature - difficulties of such an attitude in sixteenth century - lack of pragmatic support for it - primary and secondary qualities in neo-Platonic tradition - consequent gradual abandonment of claims of intuition - but necessity in Renaissance of a Pythagorean metaphysic for defending alternative doctrine - the new Reality of Renaissance Science - its standards of value - particular dangers, as evidenced by Dee, of theory of world as an exemplification of mathematical truth.

XII. Divinity as the head of the hierarchy of sciences - consonance of Platonists and Christian doctrine - views on the nature of God and the soul inseparable from Renaissance scientific theory - consequent place of qualitative experience in Dee's mathematics - the Renaissance synthesis and its multivalent appeal - rational, mathematical, aesthetic, theological - its intention to be wholly adequate to all the capacities of man.