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Her book was a model of mathematical typography.  The two volume work opens with simple algebra and continues for 1,020 pages of increasingly difficult propositions, 49 pages of illustrations, and one page of "Errori" with "Correzioni."

The Ending
A Note for Scholars on the Closing
John Colson did indeed translate Agnesi’s final statement as an “ADVERTISEMENT” near the end of the second bound volume. However, he also added problems of his own, an index and errata, making her translated ending a bit obscure.   We have checked his 1801 publication in copies at Princeton and Yale libraries. 

Agnesi’s ending is very colorful, but difficult to understand in its literal meaning.  We will provide a variety of translations based on the Huntington Library copy of Instituzioni, Vol. II, p. 1020 and the Princeton copy, Vol. II, p. 340.

Paraphrased
"I will finish the Instituzioni with a warning.  The expert analyst should be industrious in trying to search for solutions to these problems and will be much more advanced by means of the techniques that are "born" during this process.

These techniques are used by illustrious mathematicians.  Thus, now it may be possible for you to read problems in other works, and find solutions in other books, that utilize the skills and techniques learned here."

END

Translation from the “Milanese” Dialect into Modern Italian
“Concludo l'opera invitando l'analista accorto a studiare i metodi che si vedono adoperati da illustri Matematici ne' Problemi delle Curve Elastiche, Catenarie, Velarie, in quello degli Isoperimetri, ed in altri, le soluzioni de' quali pubblicate s negli Atti di Lipsia, come in altre opere, si potranno leggere, a fine di acquistare quella avvedutezza e destrezza, che  necessaria.”

......and then from the Modern Italian into English.
“I will conclude this work inviting the prudent analyst to study the methods employed by notable mathematicians in solving problems regarding elastic curves, catenaria (cosh x), velaria, iso-perimeter, and others published in the Leipzig Acta, and in other works, that can be read to gain the additional shrewdness and dexterousness that is needed.”

Explanation of the Curves
The reader is called upon to remember that Agnesi  was a younger contemporary of James Bernoulli.  She undoubtedly knew of his interest in isochronous curves and his work on the construction of the catenary.  The other curves mentioned in Agnesi’s closing are more obscure.  By “elastiche” she probably meant Bernoulli’s elastic rod, fixed at one end, but stretched in another direction.  The word “velaria” probably means a sail filled with wind.  Bernoulli had even offered a prize for the general solution of isoperimetric figures, i.e., figures of a given species and given perimeter having a maximum area.

Angel J. di Billio
Jasmin Faraone Mennella
Shirley B. Gray
Pasadena, CA
April, 2005

Colson's 1801 Translation

ADVERTISEMENT
"I shall finish these Institutions with an Advertisement, which is this; that the ingenious Analyst must endeavor, with all his skill, in the solution of Problems, to avoid Second fluxions, and much more those of a higher order; and this by means of various expedients, which will offer themselves commodiously on the spot. Such artifices may be seen, as they are made use of by famous Mathematicians, in the Problems of the Elastic Curves, the Catenaria, the Velaria, in that of Isoperimetral Curves, and in others of this kind; the solutions of which may be seen in the Leipsic Acts, and other works of this nature; by which a learner may acquire such skill and dexterity, as will be very beneficial to him.


.....from one of the first pages.

Math students will recognize an
18th century version of the 
distributive property.  Algebra
notation has evolved slowly over
centuries of time.

 MGA wrote "for her own enjoyment and to help her brother."  While most of her contemporaries were publishing in Latin, she chose to publish in an Italian written dialect, a more or less purified Tuscan.  Purified Tuscan was a written language created by Dante and others, but little spoken until almost 100 years after Agnesi's death.  Her native dialect was Milanese.  MGA enjoyed learning and speaking different languages.

With wide margins, large fonts, and many illustrations, the book was easy to read.   In order to closely supervise its production and design, she had the local print shop move the presses into her father's house where she worked while surrounded by 20 younger brothers and sisters.