When Socrates spoke of the good life. (This is a fragment. For it to make sense, it must be connected in a sentence to a main idea which, in turn, may be qualified or expanded by one or more dependent clauses. See below.)

When Socrates spoke of the good life, he was referring to his doctrine that knowledge is virtue, and that its possession leads a person to virtuous deeds. (This "Socrates" clause now makes sense because it has been attached to a main idea, he was referring to his doctrine whose meaning, in turn, is expanded by two dependent clauses.)

Across the Alps on elephants. (Fragment)

Hannibal travelled across the Alps on elephants. (Complete idea)

Taking the patient’s pulse (Fragment)

The nurse is taking the patient’s pulse. (Complete idea)

During a terrorist alert (Fragment)

People race for safety during a terrorist alert. (Complete idea)

The experiment was completed in June (main idea expressed in a independent clause) after the researcher had submitted the last of the data. (dependent clause related to the main idea)

On correcting the tests, the lack of student knowledge was apparent.
(The opening phrase is dangling; it needs to relate to a subject.)

it could become

After ("after" is preferable in this context to "on".) correcting the tests, the marker was aware of the lack of student knowledge. (The opening phrase which was dangling, is now relating to a participant, i.e., "the marker". The verb, "was apparent" has been replaced with a more direct verb, "was aware".)

Examining the second stanza of the poem, the rhyme scheme is even more complex. (Again, the opening phrase is dangling; it needs to relate to a subject.)

It could become

An examination of the second stanza of the poem reveals an even more complex rhyme scheme. Here, by nomalising the floating participle, "examining" , that is, turning it into an abstract noun, the writer is able to supply a subject and gather up the meaning of the second half of the sentence with more accuracy of meaning by using the verb, "reveals".)

BUT to be even more concise and direct the writer could say:
The second stanza of the poem has an even more complex rhythm.
(This rendition is preferable to the preceding one.)

In developing countries, in job selection, preference is given to the more highly educated person. No consideration is given to efficient job performance. Students are motivated to acquire extra education. They can now compete better in the rat race. Their education does not make them more productive. It has no ultimate influence on total output or total employment.

(Here there is a string of unconnected simple sentences which cut up the message the writer is striving to give. In order to ensure that the reader comes away with the full message, the writer needs to introduce some compounding and some complexity into the sentence structure and thus enable some ideas which are closely related to be grasped immediately by the reader.)

It could become

In developing countries, in job selection, preference is given to the more highly educated person and no consideration is given to efficient job performance. Consequently, students are motivated to acquire extra education so that they can now compete better in the rat race. Since their education does not make them more productive, it has no ultimate influence on total output or total employment.

The paragraph is now more cohesive with:

• The creation of a compound sentence using "and" which allows the writer to make a definite connection between "preference" and "consideration".
• The construction of two complex sentences, the first expressing the result of such choices, the second giving the reason why higher qualifications do not influence the employment situation.)
• The use of a connective, "consequently" to carry the compound sentence forward.

The writer has now achieved a cohesive whole. Ideas flow into one another because there is variation in sentence structure and logical connections.)

The results of the research were supported by the panel who worked on an earlier project and [who] understood the difficulties of the exercise.

(The two clauses in italics match each other in construction and so emphasise the reason for the panel’s support.)

Since the Government’s decision to introduce the GST into the Australian economy, people seem to have experienced a great deal of financial anxiety.

(The clause in italics is the main idea and is placed as a climax at the end of the sentence.)

Political freedom is possible only where justice is in the seat of authority, where all orders and degrees work in harmony with the organic laws which humans neither made nor can alter.

(The main clause is placed at the beginning of the sentence and is then followed by qualifications.)

Having established the cause for the retention rates of students in the Australian History course, the researcher carefully prepared an analytical report which accounted for the favourable trends and which would assist other departments seeking to attract and maintain numbers of students.

(The main clause, written in italics, is embedded in the compound/complex sentence and surrounded by a qualifying phrase and clauses.)

The results of the test, whatever else they may indicate, point to a lack of quality input.

(The splitting of the main idea (which is in italics) and the insertion of the "whatever" clause have produced a convoluted sentence. The occasional use of this type of sentence structure is a good way of achieving variety and emphasis but should not be overused in a piece of writing See the description of convoluted writing style at the beginning of this section.)

Note that interesting writing flows from the writer who knows how to vary sentence length and structure.