Most useful are the Three infinitives

In LLPSI Cap. XXII Orberg introduces supine endings and says in the accompanying student’s manual ‘Latine Disco’ the following: The supine endings -um and -u are added to a modifie stem-form, the so called supine stem, which is also used to form the perfect participle - and the future participle. The supine stem is regularly formed by the addition of t to the present stem, e.g. saluta-: salutat-; audi-: audit-; dic-: dict-; in e-stems e is changed to i, e.g. terre: territ-; and there are several other irregularities, especially in 3rd conjugation verbs , here the addition of t may cause changes by assimilation, e.g. scrib-: script- (p is voiceless like t), claud-: claus- (dt > tt> ss> s).

When you know the three verbal stems, (1)the present stem, (2) the perfect stem and (3) the supine stem, you can derive all forms of hte verb from them. Consequently, to be able to conjgate (i.e. inflect) a Latin verb it is sufficient to know three forms, or ‘principle parts’, in which these stems are contained. Most useful are the three infinitives:

  1. The present infinitive active, e.g. scribere
  2. The perfect infinitive active, e.g. scripsisse
  3. The perfect infinitive passive, e.g. scriptum esse

These are the forms of irregular verbs that will be given in the margin when ever needed (the 3rd form will be without esse, or missing if the verb has no passive, e.g. posse potuisse;…

Now this is a bit confusing. Is he saying that the form of irregular verbs given in most dictionaries and, indeed, by himself up until now is actually of no use and that 1, 2, 3. above is what you need?? Also true to his word he provides in the margin: vincire, vinxisse, vinctum for where any dictionary / 501 Latin verbs etc will give you vincio, vincire, vinxi, vinctum.

So my question is - er - what on earth is he on about? Is he saying that if I don’t have vincire, vinxisse, victum but do have vincio, vincire, vinxi, victum that I will not be able to form the supine… I don’t understand.

Not at all. He is advancing an alternative method of learning the principal parts of verb.
Minimé. Alternatum modum proponit ad partes principales verborum discendas.

It’s also not really correct, as far as I can tell, that you can derive all the forms of the verb with just those three forms. The example given is “scrībere, scrīpsisse, scrīptum esse”, but this doesn’t tell you whether the present form is scrībō or scrībiō.

It’s not “scribiō” because it’s not “scribīre”. But, as you imply, there are some 3rd-conjugation verbs in “io” (such as “capio”) with present infinitives in “-ere”. I imagine he wants them considered as exceptional and they do behave strangely sometimes by dropping that “i”.
“Scribiō” non est quià non est “scribīre” (nec “scribeo” quià non “scribēre”). Sunt autem (ut denotis, ut “capio” exempli gratiâ) tertiae declinationis per “-io” terminata quorum per “-ere” est infinitivitus modus temporis praesentis. Nostrum Orbergensem illa exceptiones habere credo. Inusitatè quidem nonnunquàm agunt in “i” litterae omittendo.

Multas gratias vobis ago.

My point, though, is you can’t distinguish the two third-conjugation forms from the three forms alone, whereas you can if you have the four principle parts that are found in most dictionaries. So using four parts gives you an advantage that using three does not.

How many -iō third-conjugation verbs are there? It seems odd to treat them as an irregularity rather than a normal class of verbs.

One hundred and sixty nine (a significant portion of which are compounds of “facere”) out of 2122 3rd-conjugation verbs, I estimate.
Apud me, centum sexaginta novem (quorum ponderosa pars cum “facio” composita) è duo millia centum viginti duo verba declinationis tertiae.

Have you seen this dedicated forum to LLPSI?
http://discourse.textkit.com/t/lingua-latina-per-se-illustrata-by-h-orberg/10138/1