While the child is learning French and Latin he is also beginning on other subjects, many of which are neglected in the schools. The first subject to teach is simple geography (i.e. where the major bodies of land and water are on the globe). This is the first subject because it only involves the eyes and the memory. Once the natural parts of the globe are memorized, the child should move on to arithmetic. Arithmetic is the most basic sort of abstract reasoning and provides a good introduction to this sort of thought. Once addition and subtraction have been mastered the child returns to geography and learns about poles, zones, latitude, and longitude (all the necessary components for reading a map). When he has the terrestrial globe under his belt, he can turn to the celestial globe and study the constellations. From there it is a natural step to teach him astronomy, in particular the Copernican system. Once the child is fully acquainted with all sorts of globes he can move onto geometry. Here Locke thinks that the schools teach too much; just the first six books of Euclid, Locke says, are enough for a gentleman's education. After geometry the child should study chronology so that he has an idea of the entire current of time. Chronology, together with geography, prepares the child for the most important subject, history, which he should learn as soon as chronology has been mastered.
Locke does not think that children should have to study too many works on ethics, since they have been learning virtue all along (the Bible and some Tully should be sufficient). However, he thinks it is of crucial importance that they learn the laws of the country so that they can see how to best be of service. First they should study the English constitution and the ancient books of the common law, and then they should study more modern writers. In this way they will gain insight into the reasons for laws and an appreciation of their importance.
Locke thinks that rhetoric and logic, which receive so much attention in the schools, can be largely ignored. Rhetoric does not effectively teach good speaking nor does logic effectively teach good reasoning. The best way to teach a child to speak well is to have him regularly tell stories, and then later in his development to write them down. The best way to teach a child good reasoning is to expose him to examples of well-reasoned arguments; Locke cites Chillingworth's book as a prominent example. The way that logic is taught in the schools, in fact, is actually counterproductive, Locke thinks. The method used in the schools is disputation, which is basically the attempt to try to argue with everything and to never concede a point. With this as the goal, Locke points out, a child will learn no love for the truth, only for cleverness.
The last subject a child needs to learn is natural philosophy. There are two parts to natural philosophy. There is physics, which is the study of bodies or matter, and there is metaphysics, which is the study of the immaterial world (spirits, soul, God). Because our senses only report to us about the physical world people tend to believe that the physical world is all that exists (that is, they are materialists). Matter and motion, however, cannot possibly account for all of the phenomena we observe around us (as an example of such a phenomena that cannot simply be explained as the result of the laws of matter and motion, he cites Newton's recent discovery of gravitation); there must be more to the world. The materialists, therefore, are committing a grave folly. In order to make sure that a child does not fall into this folly Locke suggests that before he be exposed to physics, he carefully study the bible, so that he will come to believe in spirits. Once he already has a belief in spirits, he can begin to study physics.
Locke is not very optimistic about natural philosophy as a discipline; he claims that it can never be made into a science. Nonetheless he believes that there is much useful and even necessary knowledge to be had in the field, and still more knowledge that is simply a delight to study. However, he thinks that any advantage to be found in natural science can be found in the experimental, observational work, and not in the speculative work. He suggests introducing the child to a little bit of every school of thought so that he can converse widely on the subject.
Locke tells us in this section that the proper way to teach a subject is to introduce one simple idea at a time. Once the child has fully grasped that simple idea, you can then introduce another simple idea that has a clear logical connection to the first. Though he does not give an example, arithmetic would provide an excellent one. To teach a child to count, you can first introduce him to the concept of a single thing. Once he has fully mastered the idea of "one", you can introduce him to the idea of "two", by putting two "ones" together. The idea of "one" is one simple idea, the idea of "two" is another simple idea, with an obvious attachment to the first.
Though he never states this explicitly, it is clear that the method he advocates for the teaching of each individual subject, parallels the method by which he chooses his entire course of study. Each subject is taught separately (in simple parts) and is followed by a subject that bears an obvious logical connection to it. With his carefully planned out curriculum, Locke aims to present a rational pattern to knowledge and to tailor academic learning to the developing mind of the child.
Locke claims that natural philosophy can never be a science. He means that we can never really have a systematic body of knowledge in natural philosophy. Locke is working with a very strict definition of knowledge here. Knowledge is the perception of a connection (either agreement or disagreement) between two or more ideas. The connection that needs to exist between ideas in order for them to count as knowledge is very strong. In the case of disagreement, the connection must be one of logical inconsistency. A square circle is an example of two logically inconsistent ideas. A married bachelor is another such example. In the case of agreement between ideas, the connection needs to be a necessary connection. That is to say, in order to know that A caused B you need to know that given A, B could not have failed to happen. Another way to put this is to say that in order to know that A caused B, you need to be able to deduce B given only the information that A, or derive B from A. As an example, consider one ball hitting another and causing the other one to move. In order to know that the first ball caused to second ball to move, you must know that the second ball could not have failed to move given the fact that the first one hit it. Or, to put it the other way, in order to know that the first ball caused the second ball to move, it has to have been possible for you to have predicted with certainty that the second ball would move, as soon as you knew that the first ball hit it.
Given this strict definition of knowledge, Locke does not think that we can have any knowledge concerning natural philosophy (that is, we cannot make it into a science). All that we can do is go through the world and observe certain qualities regularly co-occurring. We can see, for instance, that gold is malleable, yellow, fusible, soluble in aqua regia, etc. This, however, does not give us knowledge of the nature of gold because we do not see any necessary connections that would explain why gold has all of these properties regularly co-occurring. We do not see any necessary co-existence between these properties. The kind of connection that Locke is demanding is the sort that we find between properties regularly co-occurring in geometrical figures. In those cases, we can deduce the properties and see why they are necessarily co-existent. For instance, if we want to know why the angles of a triangle always add up to 180 degrees, we can construct a mathematical proof that shows us why this is necessarily the case (that is, why it could not possibly have been otherwise).
Locke does consider the possibility that we could find a necessary connection between the observable properties and the microstructure of the objects they belong to. At IV.iii.11 of the Essay he states explicitly that if we had access to the microstructures (say, with a very powerful microscope) we would be able to deduce from it the observable qualities to which it gives rise. In other words, we would see the necessary connection between the microstructure and the observable qualities, and would therefore have knowledge of the nature of things. In section thirteen, however, he reigns this fleeting optimism in. Even if we did gain access to the microstructures, he tells us, there would still be an insuperable obstacle to our knowledge. The problem is that while there is a necessary connection between the microstructure and the primary qualities we experience (i.e. shape, number, texture), there is no necessary connection between the microstructure and the secondary qualities that we experience (i.e. color, sound, taste, smell, feel). There is no reason, Locke claims, why such and such an arrangement of matter should give rise to the sensation of sweetness or of blue. It is simply God's arbitrary decision that forges these connections. God could easily have set things up differently, so that, for instance, the microstructure that now gives rise to our sensation of yellow could actually give rise to the sensation of blue or even to the smell of chocolate. Given that a large percentage of what we observe about the world is secondary qualities, this is a pretty considerable obstacle to knowledge.